Հավանականությունների տեսության խնդրագրիք

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

Ն. Գ. ԱՀԱՐՈՆՅԱՆ | Ե. Ռ. ԻՍՐԱԵԼՅԱՆ

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ

ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆ

Ն. Գ. ԱՀԱՐՈՆՅԱՆ

Ե. Ռ. ԻՍՐԱԵԼՅԱՆ

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ

ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

ԵՐԵՎԱՆ

ԵՊՀ ՀՐԱՏԱՐԱԿՉՈՒԹՅՈՒՆ

ՀՏԴ 519.2(076.1) ԳՄԴ 22.171ց7 Ա 420

Հրատարակության է երաշխավորել ԵՊՀ մաթեմատիկայի և մեխանիկայի ֆակուլտետի գիտական խորհուրդը

Խմբագիր` Գրախոսներ`

Ա 420

ֆիզ-մաթ. գիտ. դոկտոր, պրոֆեսոր

Վ. Կ. ՕՀԱՆՅԱՆ

ֆիզ-մաթ. գիտ. թեկն., դոցենտ

Ս. Մ. ՆԱՐԻՄԱՆՅԱՆ

ֆիզ-մաթ. գիտ. թեկն., դոցենտ

Կ. Վ. ԳԱՍՊԱՐՅԱՆ

Ահարոնյան Ն. Գ., Իսրաելյան Ե. Ռ. Հավանականությունների տեսության խնդրագիրք/ Ն. Գ. Ահարոնյան, Ե. Ռ. Իսրաելյան: -Եր., ԵՊՀ հրատ., 2016, 154 էջ:

Խնդրագիրքը կազմվել է Երևանի պետական համալսարանի մաթեմատիկայի և մեխանիկայի, ինֆորմատիկայի, ֆիզիկայի, ռադիոֆիզիկայի, տնտեսագիտության ֆակուլտետներում դասավանդվող «Հավանականությունների տեսություն» առարկայի ծրագրին համապատասխան: Նախատեսվում է առկա և հեռակա ուսուցման ուսանողների համար:

ՀՏԴ 519.2(076.1) ԳՄԴ 22.171ց7

ISBN 978-5-8084-2119-6  ԵՊՀ հրատ., 2016  Ն. Գ. Ահարոնյան, 2016  Ե. Ռ. Իսրաելյան, 2016

XMBAGRI KOMIC

Xndragirq kazmva maematikayi mexanikayi, infor{ matikayi kira akan maematikayi, fizikayi, adiofizi{ kayi, tntesagitowyan fakowltetnerowm dasavandvo <Hava{ nakanowyownneri tesowyown> a arkayi ragrin hamapatas{ xan: Ays xndragirq parownakowm tesakan nyowi hakir a{ radranq 645 xndir: Heinakner gtagor elov irenc dasa{ xosakan or, xndirner baanel en A B xmberi` havi a { nelov tarber fakowltetneri pahanjner: Ayd isk pata ov xndragirq karo gtakar linel inpes EPH-i, aynpes l ayl BOWH-eri ayn owsanoneri hamar, oronq owsowmnasirowm en <Ha{ vanakanowyownneri tesowyown> a arkan: Xndragirq parow{ nakowm xndirner havanakanowyownneri tesowyan ndhanowr dasnaci bolor bainneric: Ays hraparakman mej i tarbe{ rowyown a ajini (1986) erkrordi (1997), avelacvel xndirneri ndhanowr qanak: Xndirneri low man hamar gtagor vo mae{ matikakan aparat dowrs i galis maematikakan analizi, hanrahavi analitik erkraa owyan standart dasnac{ neri sahmanneric: Xndragrqi avorman nacqowm gtagor { vel en mi arq xndragrqer dasagrqer, masnavorapes Mesro{ pyan N. X., azanyan T. P., <Havanakanowyownneri tesowyan xndragirq>, mas a ajin erkrord: V. K. hanyan

Gor oowyownner patahowyneri het Havanakanowyownneri tesowyan mej ndownva het yal mo{ tecowm` yowraqanyowr orin hamapatasxanowyan mej drvowm tarrakan patahowyneri Ω bazmowyown` ori bolor irar ba{ ca o elqeri hamaxmbowyown: Ω-i enabazmowyownneri o da{ tark F das anvanowm en σ-hanrahaiv, ee bavararva en het yal paymanner` 1) ee A ∈ F , apa Ā ∈ F , orte A = Ω \ A, ∞ ∩ 2) ee A1 , A2 , . . . An , . . . ∈ F , apa An ∈ F : n=1 Tvyal ori het kapva patahowyner Ω tarrakan pata{ howyneri bazmowyan enabazmowyownner en, oronq patkanowm en F -in: Aysowhet patahowyner kanvanenq miayn F -i tarrer: Patahowyner nanakowm en A, B, C, ... ta erov: Aknhayt , or ∅, Ω ∈ F : Ω patahowy anvanowm en havasti patahowy, ∅ patahowy anvanowm en anhnar patahowy: A patahowy ko{ vowm B patahowyi masnavor depq (A ⊂ B), ee A patahowyi handes galowc het owm B patahowyi handes gal: Masnavo{ rapes, A ⊂ A ∅ ⊂ A cankaca A-i hamar: A B patahowy{ ner hamnknowm en, ee A ⊂ B B ⊂ A: A B patahowyneri A ∪ B miavorowm aynpisi patahowy , or tei ownenowm, erb handes galis A B patahowyneric gone mek: A B pa{ tahowyneri A ∩ B hatowm patahowy , or tei ownenowm, erb A-n B -n handes en galis hamate: A B patahowyneri A r B tarberowyown patahowy , or tei ownenowm, erb tei ownenowm A-n, bayc tei i ownenowm B -n: A B patahowyneri A △ B simetrik tarberowyown sahmanvowm orpes (A r B) ∪ (B r A): A B patahowyner kovowm en anhamateeli, ee A ∩ B = ∅: A1 , A2 , . . . , An patahowyner kazmowm en lriv xowmb, n ∪ ee Ai ∩ Aj = ∅, i ̸= j Ai = Ω: A Ā kovowm en hakadir i=1 patahowyner, ee A ∪ Ā = Ω, A ∩ Ā = ∅: A patahowyi P (A) havanakanowyown vayin fownkcia , or orova F σ-hanrahavi vra bavararowm het yal aqsiom{

nerin` 1) P (A) ≥ 0 ∀A ∈ F , 2) P (Ω) = 1, 3) ee A1 , A2 , . . . , An , . . . patahowyner zowyg a zowyg anha{ ∞ ∞ ∩ ∪ ∑ mateeli en` Ai Aj = ∅, i ̸= j , apa P ( Ai ) = P (Ai ): i=1 i=1 (Ω, F, P ) e yakn anvanowm en havanakanayin tara owyown:

1. Metaadram netvowm erkow angam: Nkaragrel tarrakan patahowyneri bazmowyown: Nkaragrel het yal patahowyne{ r` A - gone mek angam ker a gerb, B - gerb ker a erkrord netman amanak: 2. Za netowm en erkow angam: Nkaragrel tarrakan pata{ howyneri bazmowyown: Nkaragrel het yal patahowyner` A - er aco miavorneri gowmar havasar <8>-i, B - gone mek angam ker a <6>-: 3. Metaadram netvowm aynqan angam, min er a gerb: Nkaragrel het yal patahowyner` A - gone mek angam ker a gir, B - gerb ker a errord netman depqowm: 4. Metaadram netvowm aynqan angam, min nowyn kom irar et ic er a erkow angam: Nkaragrel het yal patahowy{ ner` A - katarel en it 5 netowm, B - gir ker a o avel, qan 4 angam: 5. Xanow axatowm am 9-ic min 18-: Patahakan gnord mtnowm xanow amanaki x pahin he anowm xanowic amanaki y pahin: Nkaragrel (x, y) tarrakan patahowyneri bazmowyown: x-i y-i terminnerov nkaragrel het yal pata{ howyner`

a) gnord gtnvowm xanowowm mek amic o aveli, b) amanaki z pahin gnord gtnvowm xanowowm: 6. 1, 2, . . . n veric patahakanoren vercra mek iv: Dicowq patahowy` ntra iv baanvowm ereqi, B patahowy` ntra iv zowyg : I?n en nanakowm A ∩ B , A ∪ B , A r B patahowyner: A

7. 1, 2, . . . n veric patahakanoren ntra mek iv: Dicowq patahowy` ntra iv baanvowm hingi, B patahowy` ntra iv avartvowm zroyov: I?n en nanakowm ArB A∩ B̄ patahowyner: A

8. Netowm en erkow za : Dicowq A patahowy` er aco mia{ vorneri gowmar kent , B -n` a nvazn mek za i vra ker a <1>-: Nkaragrel A ∩ B , A ∪ B patahowyner: 9. Apacowcel, or A ⊃ B depqowm tei kownenan` a) Ā ⊂ B̄ , b) A ∩ B = B , g) A ∪ B = A: 10. Apacowcel ∪ ∩ havasarowyownner` a) Ai = Ai i∈I ∩

i∈I ∪

b) Ai = Ai i∈I i∈I orte I -n indeqsneri kamayakan bazmowyown : 11. Gtnel x patahowy 12. Apacowcel, or en lriv xowmb:

(x ∪ A)∪(x ∪ Ā) = B

A, Ā ∩ B, A ∪ B

havasarowmic:

patahowyner kazmowm

13. Berel rinakner` 1) ereq patahowyneri, oronq, linelov havasarahnaravor anhamateeli, en kazmowm patahowyneri lriv xowmb, 2) ors patahowyneri, oronq, linelov havasarahnaravor, kazmowm en lriv xowmb:

14. Erkow patahowyneri A ∪ B miavorowm artahaytel anhamateeli patahowyneri miavorman mijocov: Ereq patahowyneri A ∪ B ∪ C miavorowm artahaytel an{ hamateeli patahowyneri miavorman mijocov: 15. Nel het yal patahowyneri hakadirner` a) A { metaadrami erkow netowmneri depqowm ker a gerb, b) B { ereq krakocneri depqowm nanin kdipen ereq angam, g) C { ereq krakocneri depqowm nanin kdipen gone mek an{ gam: 16. iraxin krakowm en ereq angam: Dicowq Ai -n (i = 1, 2, 3) patahowyn , erb i-rd krakoc dipowk : Nkaragrel Ai pata{ howynerov het yal patahowyner` a) A { iraxin kdipen ereq angam, b) B { iraxin o mi angam en dipi, g) C { iraxin kdipen miayn mek angam, d) D { iraxin kdipen a nvazn erkow angam: 17. Sarq bakaca a ajin tipi erkow erkrord tipi ereq maseric: Ditarkenq het yal patahowyner` Ak (k = 1, 2) - a ajin tipi k-rd mas Bj (j = 1, 2, 3) - erkrord tipi j -rd mas axatownak : Sarq anxa an , ee axatownak en a ajin tipi maseric gone mek erkrord tipi maseric gone erkows: Nkaragrel Ak Bj patahowyneri mijocov sarqi an{ xa an linelow C patahowy: 18. Dicowq

Ω = R2 , F = B2 , A = {(x, y) : x + y ≤ 1}, B = {(x, y) : y ≤ 2x + 2}: Nkaragrel A ∩ B, A ∪ B, A r B, Ā, B̄, Ā ∩ B̄

patahowyner: 19. Dicowq

1 1 Ω = R1 , F = B1 , An = ( 2n , n) : ∞ ∞ ∩ ∪ An , B = An patahowyner:

n=1

Nkaragrel

A =

B 20. Dicowq ∞ ∪

An ∈ F, n = 1, 2, . . .:

Bn ,

orte B1 = A1 , n=1 anhamateeli en:

Bn = An r

n−1 ∪

Apacowcel, or Ai

i=1

∞ ∪

An =

n=1

patahowyner

21. Apacowcel, or a) A △ B = (A ∪ B) r (A ∩ B), b) A △ (B △ C) = (A △ B) △ C , g) A △ ∅ = A, A △ Ω = Ā, d) A △ A = ∅, A △ Ā = Ω: 22. Dicowq An ∈ F, n = 1, 2, . . . , A∗ - ayn miayn ayn ω tarreri bazmowyownn , oronq patkanowm en anverj vov An patahowynerin: A∗ - ayn miayn ayn ω tarreri bazmowyownn , oronq patkanowm en bolor patahowynerin, baca owyamb ver{ javor vov patahowyneri: Apacowcel, or ∞ ∩ ∞ ∪ a) An ⊂ A∗ , n = 1, 2, ..., b) A∗ = Am , g) A∗ =

∞ ∪ ∞ ∩ n=1 m=n

n=1 m=n

Ditoowyown.

A∗ = lim An = lim sup An {An } hajordakanow{ n→∞ n→∞ sahmann , A∗ = lim An = lim inf An hajordaka{

yan verin n→∞ n→∞ nowyan storin sahmann : Kasenq, or {An } hajordakanowyown owni sahman, ee A∗ = A∗ : 23. Dicowq

{

ee ω ∈/ A ee ω ∈ A : IA (ω)-n kovowm A bazmowyan indikator fownkcia: Apacowcel, or a) IA∩B (ω) = IA (ω) · IB (ω), IA (ω) =

0, 1,

b) g) d) e) z)

IA∪B (ω) = IA (ω) + IB (ω) − IA (ω) · IB (ω), IĀ (ω) = 1 − IA (ω), IArB (ω) = IA (ω)[1 − IB (ω)], IA△B (ω) = |IA (ω) − IB (ω)| IA△B (ω) = [IA { (ω) + IB (ω)](mod 2), 0, ee a-n zowyg , orte a(mod 2) = 1, ee a-n kent :

24. Dicowq { An =

A, B,

ee n − zowyg ee n − kent :

Apacowcel, or n→∞ lim An = A ∪ B,

lim An = A ∩ B :

n→∞

25. Dicowq An ∈ F , n = 1, 2, . . . bolor n-eri hamar ∞ ∪ An+1 : Apacowcel, or lim An = lim An = An : n→∞

An ⊂

26. Dicowq An ∈ F , n = 1, 2, . . . bolor n-eri hamar ∞ ∩ An+1 : Apacowcel, or lim An = lim An = An : n→∞

An ⊃

n→∞

n=1

Hamakcowyown (Kombinatorika) 1. Hamakcowyan himnakan skzbownq (bazmapatkman skzbownq): Dicowq anhraet hajordabar katarel k gor oowyown: Ee a ajin gor oowyown kareli katarel n1 tarber erov, aynowhet erkrord` n2 , isk errord` n3 erov ayln, min k-rd gor oowyown, or kareli katarel nk tarber erov, apa bolor k gor oowyownner kareli katarel n1 · n2 · · · nk tarber erov: Dicowq ownenq n tarreric bakaca hamaxmbowyown: 2. Kargavorowyown n-ic k-akan` aynpisi miacowyownner en, oroncic yowraqanyowr parownakowm n tarreric vercra k tarr

oronq tarbervowm en mimyancic kam gone mek tarrov, kam dranc dasavorowyamb, nd orowm a anin miacowyownnerowm yowraqan{ yowr tarr masnakcowm o aveli, qan mek angam: n-ic k-akan kargavorowyownneri iv nanakvowm Akn -ov havasar Akn = n(n − 1) · · · (n − (k − 1)) =

n! (n − k)!

(kargavorva anveradar ntrowyown): 3. n tarreric tea oxowyownner aynpisi miacowyownner en, oroncic yowraqanyowr parownakowm n tarr, oronq mimyancic tarbervowm en miayn tarreri dasavorowyamb: n tarreric te{ a oxowyownneri iv nanakvowm Pn -ov havasar Pn = Ann = n!

4. Zowgordowyownner n-ic k-akan aynpisi miacowyownner en, oroncic yowraqanyowr parownakowm tvyal n tarreric vercra

k tarr oronq tarbervowm en mimyancic gone mek tarrov, nd orowm a anin miacowyownnerowm yowraqanyowr tarr masnakcowm o aveli, qan mek angam: n-ic k-akan zowgordowyownneri iv nanakvowm Cnk -ov havasar Cnk =

n! k!(n − k)!

(o kargavorva anveradar ntrowyown): 5. Kargavorva veradarowmov ntrowyown` k |n · n{z· · · n} = n : k

6. O kargavorva veradarowmov ntrowyown` n−1 k Cn+k−1 = Cn+k−1 :

A 27. Sari gaga kareli hasnel 7 owinerov: Qani? tarber erov le nagnac karo barranal ijnel saric: Low el xndi{ r, ee verelq vayrjq katarvowm en tarber owinerov: 28. Dasaranowm ancnowm en 10 a arka: Erkowabi r 6 das , nd orowm bolor daser tarber en: Qani? tarber erov kareli kazmel dasacowcak erkowabi rva hamar: 29. Owsano petq 8 rva nacqowm hanni 4 qnnowyown: Qani? tarber erov kareli da irakanacnel, ee mek rowm kareli hannel miayn mek qnnowyown: 30. Qani? ankyownagi owni ow owcik n ankyownin: 31. Qani? ov kareli dasavorel axmati taxtaki vra 8 navak, orpeszi nranq karoanan harva el mimyanc: 32. Gtnel n tarreric aynpisi tea oxowyownneri iv, or{ te tvyal 2 tarr en gtnvowm irar mot: 33. Paron Jons owni 10 girq, oronq na patrastvowm tea{ drel ir gradarakowm: Drancic 4- maematikakan grqer en, 3- verabervowm en qimiayin, 2-` patmowyan, isk 1-` angleren lezvin: Jons cankanowm dasavorel ir grqer aynpes, or nowyn a arkayin verabervo grqer drva linen miasin: Qani tarber dasavorowyownner en hnaravor: 34. In-or qaaqi ostikanowyan banowm axatowm 10 spa: Ee ostikanakan varowyown 5 spaneri petq owarki ooc{ ner hskelow, 2- petq axaten lriv drowyqov kayanowm, isk 3-` pahowstayin kayanowm, qani? tarber hnaravorowyownner kan 10 spanerin 3 xmberi baanelow hamar:

B 35. Oroel ow owcik n-ankyan ankyownag eri hatman keteri qanak, ee drancic yowraqanyowr ereq en hatvowm nowyn ketowm: 36. Hannaoov bakaca 11 hogowc: astaer, oron{ cov petq zbavi hannaoov, gtnvowm en paharanowm: Qani?

akan petq ownena paharan qani banali petq ownena hannaoovi yowraqanyowr andam, orpeszi hnaravor lini bacel paharan ayn miayn ayn depqowm, erb nerka klini hannaoovi andamneri me amasnowyown: 37. n mianman gndikner teavorowm en N sa orneri mej: Apacowcel, or −1 a) tarber teavorowmneri iv havasar CNn +n−1 = CNN+n−1 , b) teavorowmneri iv, erb yowraqanyowr sa or kparownaki N −1 a nvazn mek gndik, havasar Cn−1 : 38. Qani? tarber erov kareli baxel N erexaneri mij mianman nver: Gtnel ayn eanakneri iv, erb yowraqanyowr erexa kstana a nvazn mek nver: n

39. Qani? tarber erov hnaravor ntrel 6 karkandak, ee hrowakaranowm kan 11 tarber tesaki karkandakner: 40. a) Qani? amboj o bacasakan low owm owni x1 + x2 + . . . + xN = n

havasarowm: b) Qani? amboj drakan low owm owni x1 + x2 + . . . + xN = n

havasarowm:

41. Dicowq ownenq N o oxakanneric f (x1 , x2 , . . . , xN ) fownk{ cia: Qani? tarber n-rd kargi masnaki a ancyalner owni ayd fownkcian: 42. 10 mardkancic bakaca akowmbic, oronc vowm en A, B , C , D, E -n F -, harkavor ntrel naxagah, ganapah qartowar: O mek i karo ownenal mekic aveli paton: Pe{ takan patonyaneri qani? tarber ntrowyownner en hnaravor, ee a) o mi sahmana akowm ka, b) A-n B -n en ntrvi miasin, g) C -n D-n kam miasin kntrven, kam en ntrvi, d) E -n klini patonya, e) F - hamaayn axatel miayn naxagahi patonowm:

Havanakanowyan dasakan sahmanowm Dicowq Ω tara owyown bakaca n havasarahnaravor tarrakan patahowyneric` Ω = {ω1 , ω2 , . . . , ωn }: Cankaca A ⊂ Ω, A = {ωi1 , ωi2 , . . . , ωik }, k ≤ n patahowyi havanakanowyown havasar P (A) =

k : n

43. Metaadram netvowm erkow angam: Inpisi?n gerbi gone mek angam er alow havanakanowyown: 44. Netowm en erkow za : Gtnel het yal patahowyneri ha{ vanakanowyownner` a) za eri vra ker an mi nowyn qanaki miavorner, b) za eri vra ker an tarber qanaki miavorner: 45. He axosahamar bakaca 6 vananic: Gtnel bolor vananneri tarber linelow havanakanowyown:

46. Ow harkani enqi verelakn en mtnowm a ajin harkowm 5 hogi: Enadrenq, or nrancic yowraqanyowr havasar havana{ kanowyamb karo dowrs gal cankaca harkowm, sksa erkrord harkic: Gtnel het yal patahowyneri havanakanowyownner` a) bolor dowrs kgan mi nowyn harkowm, b) bolor dowrs kgan tarber harkerowm: 47. k hranoic bakaca martkoc krak bacel l inqnai { neric bakaca xmbi vra (k ≤ l): Hranoneric yowraqanyowrn ir npatakaketi ntrowyown katarowm patahakanoren, an{ kax myowsneric: Gtnel het yal patahowyneri havanakanow{ yown` a) bolor k hranoner ntrel en mi nowyn npatakaket, b) hranonern ntrel en tarber npatakaketer: 48. 1, 2, 3, 4, 5 ver grva en 5 qarteri vra: Patahakanoren hajordabar hanowm en ereq qart hanva ver dasavorowm en axic aj: Inpisi?n stacva vi a) zowyg, b) kent linelow havanakanowyown: 49. A anin qarteri vra grva en 1, 2, 3, ..., 9 ver: Bolor qarter xa nelowc heto patahakanoren hajordabar hanowm en nrancic ors dasavorowm mek myowsi het ic: Inpisi?n stacva vi a) zowyg, b) 1 2 3 4 linelow havanakanowyown: 50. Ark parownakowm 15 detal, oroncic 10- nerkva en: Banvor patahakan vercnowm nrancic 3-: Gtnel bolor verc{ ra detalneri nerkva linelow havanakanowyown: 51. Sa or parownakowm a spitak b s gndikner: Sa oric miangamic hanowm en erkow gndik: Oroel ayd erkow gndikneri spitak linelow havanakanowyown: 52. N detalneric bakaca xmbaqanak gtnvowm hskii mot, or patahakanoren ntrowm n detal oroowm dranc

orak: Ee ntra detalneric o mek xotanva , apa amboj xmbaqanak ndownvowm : Inpisi? havanakanowyamb hski kndowni k xotanva detal parownako xmbaqanak: 53. 20 owsanoneric bakaca xmbowm ka 6 gerazancik: Gtnel patahakanoren ntra 9 owsanoneric 4 - i gerazancik linelow havanakanowyown: 54. axmati mrcman masnakcowm 20 hogi, oronq viaka{ hanowyamb baanvowm en 10 hogowc bakaca erkow xmbi: Gtnel het yal patahowyneri havanakanowyownner` a) erkow amenaowe xaaconer kbaxven tarber xmberi mej, b) ors amenaowe xaaconer kbaxven erkowakan tarber xmberi mej: 55. Xaaeri kapowk patahakanoren baanvowm erkow havasar maseri: Oroel yowraqanyowr masowm erkowakan mekanoc linelow havanakanowyown: 56. N artadranqneric bakaca xmbaqanak parownakowm M xotanva artadranq: Ayd xmbaqanakic patahakanoren vercnowm en n (n ≤ N ) artadranq: Ini? havasar dranc mej m (m ≤ M ) xotanva artadranqner linelow havanakanowyow{ n: 57. 1 - 49 veric patahakan nvowm en 6 tarber ver: Inpi{ si? havanakanowyamb viakaxai amanak hanva 6 veri mej kgtnven nva veric gone 3-: 58. Xaarkvowm viakaxai n toms, oroncic m- aho : Inpisi?n ahelow havanakanowyown r (r < m) toms e q beroi hamar: 59. Meqenaneri 12 kanga ner dasavorva en mek arqov: Nkatvec, or kanga neric ow zbaecva , isk ors azat teer

het owm en mek myowsin (kazmowm en seria): Gtnel aydpisi dasavo{ rowyan havanakanowyown: 60. Kanga in moteco meqenan zbaecnowm arqi N teeric mek (o ayramasayin): Verada nalis meqenayi ter nkatowm , or N teeric r - de zbaecva : Gtnel erkow har an teeri azat linelow havanakanowyown: 61. Netva 10 za : Gtnel het yal patahowyneri havana{ kanowyownner` a) o mi za i vra i bacvel <6>, b) <6> bacvel it 3 za i vra: 62. 1, 2, 3, . . . , 29, 30 veric patahakan nvowm en 10 tarber ver: Gtnel het yal patahowyneri havanakanowyownner` a) bolor ntrva ver kent en, b) ntrva veric owi 5- baanvowm en 3-i, g) ntrva veric 5- zowyg en, 5-` kent, nd orowm drancic it mek baanvowm 10-i: 63. Inpisi? havanakanowyamb patahakanoren vercra av{ tomeqenayi qa ani hamari` a) bolor ver klinen tarber, b) veric miayn erkows khamnknen, g) ver kkazmen hamnkno veric bakaca erkow zowyg, d) veric ereq khamnknen, e) bolor ver khamnknen: 64. 0, 1, 2 veric bakaca n erkarowyown owneco bolor ha{ jordakanowyownneri bazmowyownic patahakanoren ntrvowm mek: Gtnel het yal patahowyneri havanakanowyownner` a) hajordakanowyown sksvowm zroyov, b) hajordakanowyown parownakowm owi m + 2 zroner, nd orowm drancic erkows gtnvowm en hajordakanowyan ayrerowm, g) hajordakanowyown parownakowm owi m miavor, d) hajordakanowyown parownakowm owi m0 hat <0>, m1 hat <1>, m2 hat <2>:

65. Dahliowm ka n + k nstaran, mardik patahakanoren zbaecnowm en n te: Gtnel havanakanowyown, or kzbaecvi oroaki it m (m ≤ n) te: 66. Ownenq dasavorva 52 xaaeri kapowk: Gtnel hava{ nakanowyown, or a) a ajin ors xaaer klinen mekanoc, b) a ajin verjin xaaer mekanoc en, g) mekanocneri mij ka it l xaaow: 67. 52 xaaeric kazmva kapowk havasar baanvowm 4 xaaconeri mij : Gtnel havanakanowyown, or a) yowraqanyowr xaacoi mot klini mekanoc, b) xaaconeric meki mot bolor 13 xaaer klinen nowyn tesaki, g) yowraqanyowr xaacoi mot klinen bolor xaaeric` erkowsic min mekanoc: 68. 15 dasagirq patahakan kargov dasavorva en daraki vra, nd orowm drancic 5- kazmov en: Aakert patahakanoren vercnowm drancic 3-: Inpisi? havanakanowyamb vercra grqe{ ric gone mek kazmov : 69. Netowm en 3 za : Gtnel <6>-i gone mek angam er alow havanakanowyown: 70. Ark parownakowm 8 karmir, 10 kana, 12 kapowyt gndik{ ner: Patahakanoren hanowm en nrancic ereq: Gtnel hanva

gndikneric gone erkowsi nowyn gowyni linelow havanakanowyown: 71. 9 ow or nstowm 3 vagonic bakaca gnacq: Yowraqan{ yowr ow or patahakan ntrowm vagonneric mek: Gtnel he{ t yal patahowyneri havanakanowyownner` a) bolor kntren a ajin vagon, b) bolor kntren mi nowyn vagon,

g) ow orneric gone mek kntri a ajin vagon, d) yowraqanyowr vagon kbarrana 3 ow or: 72. Gtnel 12 aneri nndyan reri tarva tarber amisnerowm linelow havanakanowyown: 73. Qa akowsin horizonakan g erov baanva n mianman erteri: Drancic yowraqanyowri vra patahakanoren nvowm mi ket, ori dirq havasarahnaravor erti cankaca teowm: Aynowhet ayd qa akowsin baanvowm n − 1 owaig g erov: Gtnel yowraqanyowr owaig ertowm mekakan ket gtnvelow ha{ vanakanowyown: 74. Sa oric, or parownakowm 2 spitak 4 s gndikner irar het ic hanowm en bolor gndikner: Gtnel het yal pata{ howyneri havanakanowyownner` a) a ajin hanva gndik spitak , b) erkrord hanva gndik spitak , g) verjin hanva gndik spitak : 75. Viakaxai 7 tomseric erkows aho en: 7 hogi hajor{ dabar vercnowm en mekakan toms: Kaxva klini? ardyoq ahelow havanakanowyown heri hamaric: 76. n nkerner patahakanoren nstowm en klor seani owrj: Gtnel het yal patahowyneri havanakanowyownner` a) oroaki erkows` A-n B -n, nsta en koq-koqi, b) A-n nsta B -ic ax, g) oroaki ereq` A-n, B -n C -n nsta en koq-koqi, d) A-n nsta B -ic aj, isk C -n` B -ic ax: 77. Gtnel naxord xndri patahowyneri havanakanowyown{ ner, ee nkerner nsta en arqov mek nstaranin: 78. Darakowm patahakan herakanowyamb dasavorva 40 girq, oronc vowm na S. Zoryani ereq hatorner: Gtnel het yal patahowyneri havanakanowyownner`

a) S. Zoryani hatorner dasavorva en irar mot, b) S. Zoryani hatorner dasavorva en irar mot hamarneri aman kargov, g) S. Zoryani hatorner dasavorva en hamarneri aman kargov (partadir irar mot), d) hatorneri zbaecra teer kazmowm en vabanakan pro{ gresia, ori tarberowyown havasar 7-i: 79. Enadrenq dowq mo acel eq ez harkavor he axosi hama{ ri mek vanan havaqowm eq ayn patahakanoren: Inpisi? havanakanowyamb dowq stipva klineq anel o aveli qan erkow kan: 80. 20 erexa (10 ta 10 ajik) patahakanoren xmbavor{ vowm en zowygerov: Gtnel 10 zowygeric yowraqanyowri tarber se i erexaneric bakaca linelow havanakanowyown: 81. 5 tamard 10 kin hamaxmbvowm en ereqakan: Inpisi? havanakanowyamb stacva 5 xmberic yowraqanyowrowm klini mek tamard: 82. Dicowq netowm en erkow za : Gtnel havanakanowyown, or za eri vra bacva miavorneri gowmar havasar i-i, i = 2, 3, . . . , 11, 12: 83. Anta owm aprowm 20 hyowsisayin ejerow, oroncic hingin orsacel, pitakavorel, apa azat en arakel: Oro amanak anc ayd 20 ejerowneric 4-in orsowm en: Inpisi? havanakanow{ yamb drancic 2- pitakavorva en eel: 84. A ka n banali, oroncic mekn hamapatasxanowm ko{ peqin: Inpisi havanakanowyamb kareli bacel dow k-rd or{ owm, ee a) patahakanoren vercva banalin i masnakcowm hajord

orerin,

b) arden orva banalin karo noric masnakcel hajord

orerin:

85. k masnikner patahakanoren baxvowm en n bjijneri mej (k ≤ n): Gtnel het yal patahowyneri havanakanowyownner` a) oroaki k bjijnerowm khaytnabervi mekakan masnik, b) k bjijnerowm khaytnabervi mekakan masnik:

Xndir low el het yal paymanneri depqowm` 1) masnikner tarbervowm en, mek bjjowm nka masnikneri iv i sahmana akvowm, 2) masnikner en tarbervowm, mek bjji mej nka masnikneri iv i sahmana akvowm, 3) masnikner tarbervowm en, yowraqanyowr bjji mej karo nknel mekic o aveli masnik, 4) masnikner en tarbervowm, yowraqanyowr bjji mej karo nknel mekic o aveli masnik: 86. n owsano, oronc vowm en A-n B -n, patahakanoren arq en kangnowm: Inpisi? havanakanowyamb A-i B -i mij klini it r owsano: Cowyc tal, or ee n owsanoner rjan kazmen, apa ayd havanakanowyown klini r-ic ankax havasar n−1 : 87. Inpisi? havanakanowyamb r gndakner kareli teavo{ rel n arkeri mej aynpes, or it m ark mna datark, ee gndakner en tarbervowm bolor teavorowmner havasara{ hnaravor en: 88. Inpisi? havanakanowyamb 2n gndakner kareli teavo{ rel n arkeri mej aynpes, or o mi ark datark lini, ee gndak{ ner en tarbervowm bolor teavorowmner havasarahnaravor en: 89. Xaaeri kapowkic (52 hat) patahakanoren verc{ nowm en 6 xaaow: Gtnel vercra xaaeri mej bolor te{ saki xaaeric linelow havanakanowyown:

90. 1, 2, ..., 20 ver parownako 20 qarteric patahakanoren vercnowm en mek qart: Oroel ayd qarti vra grva vi 3-i kam 4-i vra baanvelow havanakanowyown: 91. 10 e agrer dasavorva en 30 apanakneri mej: Yowra{ qanyowr e agri hamar naxatesva 3 apanak: Inpisi? havanakanowyamb patahakanoren vercra 6 apanakneri mej i gtnvi ambojakan e agir: 92. Tomsarki mot her en kangna n + m mard, oroncic n- ownen 50-akan dram, isk myowsner` 100-akan dram: Tomsi gin 50 dram : Vaa qic a aj tomsarkowm dram kar: Inpisi? havanakanowyamb gnoneric o mek stipva i lini spasel mnacord manr dramin: 93. Arkic, or parownakowm m spitak n s gndikner (m ≥ patahakanoren hanowm en irar het ic bolor gndikner: In? havanakanowyamb kga aynpisi pah, erb hanva s gndikneri qanak khavasarvi hanva spitak gndikneri qanakin: n)

94. n aytikneric yowraqanyowr baanvowm erkow masi` er{ kar kar: Stacva 2n ktorner miavorowm en n zowygeri, oron{ cic yowraqanyowr kazmowm nor < aytik>: Gtnel het yal pa{ tahowyneri havanakanowyownner` a) bolor ktorner kmianan skzbnakan kargov, b) bolor erkar ktorner kmianan kar ktorneri het: 95. Ditarkenq ax2 +bx+c = 0 qa akowsi havasarowm, orte a, b, c -n orovowm en, hamapatasxanabar, orpes za i ereq hajordakan netowmneri ardyownqner: Gtnel` a) havasarman armatneri irakan linelow, b) havasarman armatneri acional linelow havanakanow{ yownner: 96. n + 1 mardkancic mek, orin kanvanenq <naxa no>, erkow namak growm patahakanoren ntra hasceatererin, oronq

kazmowm en <a ajin serownd>, nranq irenc herin anowm en nowy{ n` a ajacnelov <erkrord serownd>: Ev <r-d serownd> kazmo mardkancic yowraqanyowr owarkowm erkow namak pataha{ kanoren ntra hasceatererin: Gtnel <naxa noi> 1, 2, ..., r hamarnerov <serowndneric> o mekin patkanelow havanakanow{ yown: 97. n + 1 bnaki owneco qaaqowm in-or mek norowyown imanowm: Na ayn haordowm a ajin handipa in, da l mek owriin aydpes arownak: Yowraqanyowr qaylowm a ajin angam norowyown imaco havasar havanakanowyamb karo haordel ayd mard{ kancic yowraqanyowrin: Inpisi? havanakanowyamb amanaki r miavorneri nacqowm` a) norowyown krkin i hasni ayn mardown, orn a ajinn imacel da, b) mardkancic o mek i krkni norowyown: 98. 1, 2, ..., N bazmowyownic patahakan vercnowm en a iv: Gtnel lim pN , orte pN - (a2 − 1) vi 10-i baanvelow hava{ N →∞ nakanowyownn : 99. Ini? havasar 1, 2, ..., N bazmowyownic patahakan vercra bnakan vi fiqsa bnakan k vi baanvelow pN hava{ nakanowyown: Gtnel lim pN : N →∞

100. 1, 2, ..., N veri bazmowyownic patahakanoren vercnowm en (veradarov) x y ver: I?nn me . p2 = P (x2 − y 2

baanvowm 2 − i),

baanvowm 3 − i) : 101. 1, 2, ..., n bazmowyan bolor ir mej katarvo artapat{ kerowmneric patahakanoren vercnowm en mek: Gtnel het yal patahowyneri havanakanowyownner` p3 = P (x2 − y 2

a) ntra artapatkerowm n tarreric yowraqanyowr ta{ nowm 1-i, b) i tarrn owni owi k naxapatker, g) i tarrn artapatkervowm j tarri vra, d) i1 , i2 , ...ik tarrer (1 ≤ i1 < i2 < ... < ik ≤ n) hamapatas{ xanabar artapatkervowm en j1 , j2 , ..., jk -i vra: 102. 1, 2, ..., n bazmowyan oxmiareq artapatkerowm ir vra anvanowm en n astiani tea oxowyown: Bolor n astiani tea oxowyownneric patahakanoren vercnowm en mek: Gtnel het yal patahowyneri havanakanowyownner` ( ) 1 2 ··· n a) ntrva nowynakan tea oxowyown E = 1 2 · · · n , b) ntra tea oxowyown` i1 , i2 , ...ik (i1 < i2 < ... < ik ) tarrer tea oxowm hamapatasxanabar j1 , j2 , ..., jk tar{ rerin, g) i tarr tea oxowm i-in, aysinqn` i → i: 103. n masnikner baxvowm en N bjijneri mej: Nanakenq µr = µ(n, N ) ayn bjijneri qanak, oroncic yowraqanyowr ba{ xelowc heto kparownaki owi r masnik: Gtnel het yal pata{ howyneri havanakanowyownner` erb n = N 2) µ0 (n, N ) = 0, erb n = N + 1 3) µ(n, N ) = 1, erb n = N + 1 :

1) µ0 (n, N ) > 0,

104. Baanowm en 52 xaaqarteric bakaca kapowk: In{ pisi?n havanakanowyown, or tasnorserord qart klini <me{ kanoc>: Inpisi? havanakanowyamb a ajin <mekanoc> dowrs kga tasnorserord qarti vra:

Erkraa akan havanakanowyownner Dicowq Ω -n n-a ani vklidesyan Rn tara owyan verjavor Lebegi a owneco enabazmowyown : F - Ω-i st Lebegi a eli enabazmowyownneri σ-hanrahaivn : Ln (·)- Lebegi a n Rn − owm: Cankaca A ∈ F hamar erkraa akan havanakanowyown sahmanvowm het yal bana ov` P (A) =

Ln (A) : Ln (Ω)

105. Owi g i yowraqanyowr 15 metri vra teavorva en ha{ katankayin akanner, 3 metr laynowyown owneco tankn nanowm ayd g in owahayac: Inpisi?n tanki payelow havanaka{ nowyown: 106. Harowyown baanva iraric 2a he avorowyan vra gtnvo zowgahe owinerov: Harowyan vra patahakanoren netvowm r (r < a) a avov metaadram: Inpisi?n nra o mi owi het hatvelow havanakanowyown: 107. axmati anverj taxtaki vra, ori qa akowsineri ko{ meri erkarowyown a , patahakanoren netvowm 2r (2r < a) tramag ov metaadram: Gtnel het yal patahowyneri hava{ nakanowyownner` a) dram ambojovin nka klini mek qa akowsow mej, b) dram khati qa akowsow mek komic o aveli: 108. Ket patahakanoren nvowm R a avov rjani ners: Gtnel ayd keti rjanin nerg va a) qa akowsow, b) kanonavor e ankyan nersowm gtnvelow havanakanowyown: 109. l erkarowyown owneco OA hatva i vra patahakanoren nvowm en erkow B C keter, nd orowm, haytni , or OB < OC :

Gtnel BC hatva yown:

hatva ic kar linelow havanakanow{

110. Inpisi? havanakanowyamb x2 + ax + b = 0 qa akowsi havasarman armatner. a) irakan en, b) drakan en, ee a b gor akicneri areqner havasarahnaravor en 0 ≤ a ≤ 1, 0 ≤ b ≤ 1 qa akowsow nersowm: 111. Gtnel x2 +2ax+b = 0 havasarman armatneri a) irakan, b) drakan linelow havanakanowyown, ee a b gor akicneri areqner havasarahnaravor en |a| ≤ 1, |b| ≤ 1 qa akowsow nersowm: 112. l erkarowyown owneco hatva i vra patahakanoren nowm en erkow ket: Gtnel nranc mij ea he avorowyan kl-ic (0 < k < 1) oqr linelow havanakanowyown: 113. Erkow nav petq ka anven nowyn navamatowycin: Na{ veri amanelow paher ankax en havasarahnaravor rva nacqowm: Inpisi? havanakanowyamb naveric mek stipva

klini spasel navamatowyci azatman, ee a ajin navi kang a nelow amanak mek am , isk erkrordin` erkow am: 114. (0,0), (0,1), (1,0), (1,1) gaganer owneco qa akowsow ners nva M (ξ, η) ket: 1. Apacowcel, or 0 ≤ x, y ≤ 1 -i hamar P (ξ < x, η < y) = P (ξ < x) · P (η < y) = x · y :

2. 0 < z < 1-i hamar gtnel a) P (|ξ − η| < z), d) P (max(ξ, η) < z), b) P (ξ · η < z), e) P ( 21 (ξ + η) < z): g) P (min(ξ, η) < z), 115. Patahakanoren vercva en erkow drakan x y ver, oroncic yowraqanyowr me 2-ic: Gtnel ayd veri xy artadrya{ li 1-ic me linelow xy qanordi 2-in gerazancelow havanaka{ nowyown:

116. Patahakanoren vercnowm en erkow drakan x y ver, oroncic yowraqanyowr me mekic: Gtnel ayd veri gowmari mekin gerazancelow, isk xy artadryali 0,09-ic oqr linelow ha{ vanakanowyown:

117. Ow or karo gtvel T1 T2 ndmijowmnerov hajordo erkow erowineri tramvayneric: Ow ori kanga in motenalow pah oroowm en T1 T2 mijakayqerowm erkow u v keter, oronq hamapatasxanabar cowyc en talis ayn amanak, ori nac{ qowm ow or petq spasi min tvyal erowow hajord tramvayi gal: Enadrelov, or u v areqner havasarahnaravor en T1 T2 mijakayqerowm, gtnel kanga in moteco ow ori o aveli qan 0 < t ≤ min(T1 , T2 ) amanak spaselow havanakanowyown: 118. Byowfoni xndir: Harowyown baanva iraric 2a he{

avorowyan vra gtnvo zowgahe owinerov: Harowyan vra patahakanoren gcowm en 2l (l < a) erkarowyown owneco ase: Gtnel dra or owi het hatvelow havanakanowyown: 119. Bertrani taramitowyown: rjani mej <patahaka{ noren> vercnowm en mi lar: Inpisi? havanakanowyamb dra erka{ rowyown kgerazanci rjanin nerg va kanonavor e ankyan ko{ mi erkarowyan: Ardyownq kaxva , e inpes ktramabanen <patahakanoren> ba : 120. l erkarowyown owneco hatva i vra patahakanoren nowm en erkow ket: Ini? havasar` a) staca ereq hatva neric e ankyown ka owcelow havanakanowyown, b) oqragowyn masi er{ karowyown 3l -in gerazancelow havanakanowyown: 121. R a avi owneco rjanag i vra patahakanoren nvowm en A, B C keter: Gtnel ABC e ankyan a) sowrankyown, b) ha{ vasarasrown linelow havanakanowyown:

122. Gtnel yowraqanyowr a-in gerazanco erkarowyown owne{ co ereq patahakanoren vercra hatva neric e ankyown ka{

owcelow havanakanowyown: 123. Hastatown aragowyamb pttvo skava aki a j gt{ nvowm 2h erkarowyown owneco hatva , or dasavorva skava aki het mi nowyn harowyan vra aynpes, or mijnaket skava aki kentroni het miacno owi owahayac hatva in: amanaki patahakan pahin rjanag ic oa oi owowyamb owm mi masnik: Gtnel masniki hatva i vra nknelow hava{ nakanowyown, ee hatva i he avorowyown skava aki kent{ ronic havasar l-i: 124. r a avi owneco gnda masnik patahakanoren ow{ aig kerpov nknowm qa akowsi bjijner owneco eq metaalar mai vra: Horizoni het mai kazmva ankyown havasar φ-i, metaalari tramagi havasar d-i, metaalareri a anc{ qayin g eri mij he avorowyown` l-i: Gtnel masniki mai mijov azat ancnelow havanakanowyown: 125. Gtnel navi payecman havanakanowyown akana a{ koc martancelow depqowm, ee akanner dasavorva en arqov iraric L he avorowyan vra, isk navi owowyown akanneri g i het kazmowm α ankyown: Navi owowyan hatowm akanneri g i het havasarahnaravor cankaca ketowm: Navi laynowyown havasar b -i, isk akani tramagi ` d -i: 126. Sowzanav v aragowyamb arvowm L laynowyown owneco neowci erkarowyamb: Pahakanav katarowm mtakan oronowm, arvelov neowci laynqov v aragowyamb: Navi vra teadrva

haytnaberman gor iqi gor oowyan he avorowyown havasar r -i (r ≤ L) : Inpisi? havanakanowyamb pahakanav kbaca{ hayti sowzanav, ee sowzanavi navi owowyownneri hatowm havasarahnaravor neowci cankaca ketowm:

127. Inpisi? hastowyown petq ownena metaadram, orpes{ zi koi vra nknelow havanakanowyown havasar lini 31 : 128. Avtobowsi kanga in yowraqanyowr ors ropen mek motenowm A g i avtobows yowraqanyowr vec ropen mek` B g i avtobows: A g i avtobowsi B g i amenamot avtobowsi kanga in mo{ tenalow paheri mij amanakahatva havasarahnaravor 0-ic min 4 rope: Gtnel het yal patahowyneri havanakanow{ yownner` a) a ajin moteco avtobows A g i , b) erkow ropei nacqowm kmotena or g i avtobows: 129. Gtnel ax2 +bx+c = 0 qa akowsi havasarman armatneri irakan linelow havanakanowyown, ee a, b, c gor akicneri ar{ eqner havasarahnaravor en 0 < a ≤ 1, 0 < b ≤ 1, 0 < c ≤ 1 xoranardowm:

Paymanakan havanakanowyown Patahowyneri oreri ankaxowyown Dicowq

(Ω, F, P )-n havanakanayin tara owyown , B ∈ F P (B) > 0: A patahowyi paymanakan havanakanowyown B

patahowyi irakanacman paymanowm orovowm het yal bana{ ov` P (A/B) =

P (A ∩ B) : P (B)

Ayd havasarowyown kareli grel het yal tesqov` P (A ∩ B) = P (B) · P (A/B) :

Verjin bana i ndhanracowmn

P (A1 ∩ A2 ∩ . . . ∩ An ) = = P (A1 )·P (A2 /A1 )·P (A3 /A1 ∩A2 ) · · · P (An /A1 ∩A2 ∩. . .∩An−1 )

bana : A

patahowyner ankax en, ee P (A ∩ B) = P (A) · P (B) :

A1 , A2 , . . . , An patahowyner ankax en st hamaxmbowyan, ee P (Ai1 ∩ Ai2 ∩ . . . ∩ Aik ) = P (Ai1 )P (Ai2 ) · · · P (Aik ) cankaca k = 2, 3, ..., n 1 ≤ i1 < i2 < . . . < ik ≤ n : Cankaca A B patahowyneri hamar tei owni het yal

bana `

P (A ∪ B) = P (A) + P (B) − P (A ∩ B) :

ndhanowr depqowm` cankaca A1 , A2 , . . . , An patahowyneri ha{ mar (n( ≥ 2) ) ∑ ∑ P (Ai ∩ Aj ∩ Ak )− P (Ai ) − P (Ai ∩ Aj ) + i=1 i>j i=1 i>j>k (n ) ∩ n+1 . . . + (−1) P Ai , P

n ∪

n ∑

i=1

or kovowm kcman artaqsman bana : Enadrenq, or yowra{ qanyowr elq stacvowm a anin ori nacqowm: Ee a anin

orin verabero cankaca patahowy ankax myows orerin verabero cankaca patahowyic, apa kasenq, or ownenq ankax

oreri hajordakanowyown: Ditarkenq erkow kamayakan G1 G2 orer nanakenq (Ω1 , F1 , P1 ) (Ω2 , F2 , P2 ) dranc hamapatasxano havanaka{ nayin tara owyownner: Ditarkenq na (Ω, F, P ) havanaka{ nayin tara owyan <bard> or, orte Ω = Ω1 ×Ω2 -n Ω1 -i Ω2 − i dekartyan artadryaln , isk F - minimal σ-hanrahaivn ` a ajaca F1 ×F2 = {B1 ×B2 : B1 ∈ F1 , B2 ∈ F2 } owankyownneri kisahanrahavic: Asowm en, or G1 G2 orer ankax en, ee cankaca B1 ∈ F1 , B2 ∈ F2 hamar tei owni P (B1 × B2 ) = P1 (B1 ) · P2 (B2 ) = P (B1 × Ω2 ) · P (B2 × Ω1 ) :

oreri ankaxowyown sahmanvowm nman ov het yal havasarowyan mijocov`

G1 , G 2 , . . . , Gn

P (B1 × B2 × . . . × Bn ) = P1 (B1 ) · P2 (B2 ) · . . . · Pn (Bn ),

orte Bk ∈ Fk , (Ωk , Fk , Pk )-n Gk , k = 1, 2, . . . , n orin hama{ patasxano havanakanayin tara owyownn :

130. Netowm en erkow za : Gtnel erkowsi vra l <5> bacvelow paymanakan havanakanowyown, ee haytni , or bacva mia{ vorneri gowmar baanvowm 5-i: 131. Netowm en erkow za : Gtnel a nvazn mek angam <6> bac{ velow havanakanowyown, ee haytni , or bacva miavorneri gowmar havasar 8-i: 132. Netowm en ereq za : Inpisi?n drancic a nvazn meki vra <6> bacvelow havanakanowyown, ee za eri vra bacvel en tarber nister: 133. Apacowcel, or P (A) ̸= 0:

P (B/A) > P (B),

P (A/B) > P (A),

B̄) 134. Apacowcel, or P (B/A) ≥ 1 − PP ((A) , orte P (A) ̸= 0:

135. Apacowcel, or ee A P (A ∪ B) ̸= 0, apa

P (A/A ∪ B) =

patahowyner anhamateeli

P (A) : P (A) + P (B)

136. Dicowq A B patahowyner ankax en cowcel, or P (A) = 0 kam P (B) = 1:

A ⊂ B:

Apa{

137. Apacowcel, or ee A patahowyn ankax irenic, apa P (A) = 0 kam P (A) = 1: 138. Dicowq P (B/Ā) = P (B/A), P (A) ̸= 0, P (A) ̸= 0: Apacowcel, or A-n B -n ankax en:

139.

A P (B) ̸= 0:

patahowyner anhamateeli en, Kaxya?l en ardyoq ayd patahowyner: B

P (A) ̸= 0

140. A B patahowyner ankax en: Kaxya?l en ardyoq he{ t yal patahowyner` a) A B̄ , b) Ā B̄ : 141. Apacowcel, or ee A patahowy ankax B1 B2 an{ hamateeli patahowyneric, apa A-n B1 ∪ B2 ankax en: 142. Netowm en erkow za : Ditarkenq het yal patahowyner` ^a ajin za i vra kbacven zowyg vov miavorner_, ^erkrord za i vra kbacven kent vov miavorner_, ^bacva miavorneri gowmar kent _: Apacowcel, or A1 , A2 , A3 - zowyg a zowyg ankax en, bayc st hamaxmbowyan ankax en:

A1 = A2 = A3 =

143. Netowm en erkow za : Xi -n i -rd za i vra er aco miavor{ neri ivn (i = 1, 2): Ditarkenq het yal patahowyner` A1 A2 A3 A4 A5 A6

= = = = = =

^X1 - baanvowm 2-i, X2 - baanvowm 3-i_, ^X1 - baanvowm 3-i, X2 - baanvowm 2-i_, ^X1 - baanvowm X2 -i_: ^X2 - baanvowm X1 -i_, ^X1 + X2 - baanvowm 2-i_, ^X1 + X2 - baanvowm 3-i_:

Gtnel ankax patahowyneri bolor zowyger e yakner: 144. A ajin sa or parownakowm 5 spitak, 11 s 8 karmir gndikner, isk erkrord` 10 spitak, 8 s 6 karmir: Yowraqan{ yowr sa oric hanowm en mekakan gndik: Gtnel hanva gndikneri mi nowyn gowyni linelow havanakanowyown: 145. Sa or parownakowm 2 spitak, 3 s 5 karmir gndikner: Patahakanoren hanowm en 3 gndik: Gtnel drancic gone erkowsi tarber gowyni linelow havanakanowyown:

146. Erkow hraig, oronc hamar iraxin dipelow havanaka{ nowyownner hamapatasxanabar havasar en 0,7 0,8, talis en mekakan krakoc: Gtnel iraxin a) a nvazn mek angam dip{ elow, b) miayn mek angam dipelow havanakanowyown: 147. In-or mek mo acel he axosi hamari verjin vanan havaqowm ayn patahakanoren: a) Inpisi? havanakanow{ yamb na stipva klini zangel o avel qan ereq angam, b) inpe?s k oxvi havanakanowyown, ee haytni liner, or verjin vanan kent : 148. ragri 25 harceric owsano giti miayn 20-: Inpisi? havanakanowyamb na kpatasxani iren a ajarka ereq har{ cerin: 149. Viakaxai mek tomsov ahelow havanakanowyown ha{ vasar 0,8: Inpisi?n 2 toms ownecoi ahelow havanakanow{ yown: 150. Viakaxai n tomseric l- aho : Omn e q berowm k toms: Inpisi?n nrancic gone meki aho linelow havanakanow{ yown: 151. Apacowcel, or A B anhamateeli patahowyneri hamar havanakanowyown, or ankax orerowm A patahowy kiraka{ nana B -ic a aj, havasar P (A) : P (A) + P (B)

152. Dicowq A1 ∩ A2 ⊂ A: Apacowcel, or P (A) ≥ P (A1 ) + P (A2 ) − 1 :

153. rjanin nerg va qa akowsi: Inpisi? havanakanow{ yamb rjani mej patahakanoren netva hing keteric mek

kgtnvi qa akowsow nersowm, isk myowsner` mekakan yowraqanyowr segmentowm: 154. lektrakan ayi xzowm karo tei ownenal ki , i = 1, 2, 3 tarreric meki arqic dowrs galow pata ov: Tarrer ar{ qic dowrs en galis iraric ankax: Nanakenq pi -ov, i = 1, 2, 3, i-rd tarri arqic dowrs galow havanakanowyown` p1 = 0, 3, p2 = p3 = 0, 2: Gtnel ayi xzman havanakanowyown, ee tarrer miacva en a) hajordabar, b) zowgahe , g) k1 - hajordabar, isk k2 k3 zowgahe , d) a ajin erkow tarrer zowgahe errordi het:

155. Erkow xaaco hajordabar netowm en metaadram: a{ howm ayn xaaco, owm mot aveli owt kbacvi gerb: Gtnel ahelow havanakanowyown yowraqanyowr xaacoi hamar: 156. Ereq xaaco hajordabar netowm en metaadram: a{ howm ayn xaaco, owm mot aveli owt kbacvi gerb: Gtnel ahelow havanakanowyown yowraqanyowr xaacoi hamar: 157. Sa or parownakowm a spitak b s gndikner: Xai erkow masnakicner hajordabar hanowm en sa oric mekakan gn{ dik, yowraqanyowr angam veradarnelov ayn et: ahowm ayn xaaco, or aveli owt hanowm spitak gndik: Gtnel ahelow havanakanowyown yowraqanyowr xaacoi hamar: 158. Erkow hraig hajordabar krakowm en iraxin min a a{ jin dipowk krakoc: iraxin dipelow havanakanowyown a a{ jin hraigi hamar havasar 0,2-i, isk erkrordi hamar` 0,3-i: Inpisi? havanakanowyamb a ajin hraig kkatari aveli at krakoc, qan erkrord: 159. Qnnowyown hajo hannelow havanakanowyown ereq owsa{ noneric yowraqanyowri hamar hamapatasxanabar havasar 1/5, 1/4 1/3: Havel havanakanowyown, or qnnowyown hajo khannen ayd 3 owsanoneric gone erkows:

160. Sa or parownakowm 12 gndik, oroncic 4- spitak en: Ereq xaaconer hajordabar a anc veradari hanowm en mekakan gndik: Haowm ayn xaaco, ov aveli owt khani spitak gndik: Gtnel ahelow havanakanowyown yowraqanyowr xaacoi hamar: 161. agavor 2 erexaneric bakaca ntaniqic : Inpi{ si? havanakanowyamb myows erexan nra qowyrn : 162. Oroaki hamaynqi ntaniqneri 36 tokos own pahowm, own owneco ntaniqneri 22 tokos pahowm na katow: Baci ayd, ayd hamaynqi ntaniqneri 30 tokos pahowm katow: Ini? havasar. a) havanakanowyown, or patahakanoren ntrva nta{ niq owni ' own, ' katow, b) paymanakan havanakanowyown, or patahakanoren n{ trva ntaniq pahowm own, ee ayn pahowm katow: 163. Oroaki qoleji owsanoneri 52 tokos igakan se i ner{ kayacowciner en: Ays qoleji owsanoneri hing tokos masnagi{ tanowm hamakargayin gitowyan mej: Owsanoneri erkow tokos, oronq sovorowm en hamakargayin gitowyown, ajikner en: Gtnel paymanakan havanakanowyown, or patahakanoren ntrva

owsano a) igakan se i , havi a nelov, or na masnagitanowm hamakargayin gitowyan mej, b) masnagitanowm hamakargayin gitowyan mej, havi a { nelov, or na igakan se i :

164. 00, 01, . . . , 98, 99 verov hamarakalva 100 qarteric pa{ tahakanoren ntrowm en mek: Dicowq ξ η hamapatasxanabar ntra qarti vananneri gowmarn artadryal: Gtnel P (ξ = i/η = 0):

165. Graseanowm namaki gtnvelow havanakanowyown hava{ sar p-i, nd orowm havasar havanakanowyamb ayn karo linel seani ow darakneric cankaca owm: Stowgva yo darak{ nerowm namak i haytnabervel: Inpisi? havanakanowyamb ayn kgtnvi 8-rd darakowm: 166. Apacowcel, or ee A, B, C patahowynern ankax en st hamaxmbowyan, apa a) A B ∪ C , b) A B r C ankax en: 167. Berel rinak, or cowyc talis, or P (A1 ∩ A2 ∩ A3 ) = P (A1 ) · P (A2 ) · P (A3 ) P (A3 ) > 0 paymanic i bxowm P (A1 ∩ A2 ) = P (A1 ) · P (A2 ) : 168. Netvowm erkow za : Nanakenq

Al = ^a ajin za i vra er aco miavorneri iv baanvowm l-i_, Bl = ^erkrord za i vra er aco miavorneri iv baanvowm l-i_, Cl = ^erkow za eri vra er aco miavorneri gowmar baanvowm l -i_: Anka?x en ardyoq het yal patahowyneri zowyger a) Al , Bk cankaca l- i k-i depqowm, b) A2 , C2 , g) A4 , C4 :

169. (0,0), (0,1), (1,0), (1,1) gaganer owneco qa akowsow mej patahakanoren nvowm M ket: Dicowq (ξ1 , ξ2 )- ayd keti koordinatnern en: r -i o?r areqneri depqowm Ar = (|ξ1 −ξ2 | ≥ r) Br = (ξ1 +ξ2 ≤ 3r) patahowyner klinen ankax: 170. st naxord xndri paymanneri, dicowq A1 = (ξ1 ≤ 12 ), A2 = (ξ2 ≤ 21 ), A3 = ((ξ1 − 12 )(ξ1 − 21 ) < 0): Cowyc tal or A1 , A2 , A3 patahowyner zowyg a zowyg ankax en, bayc st hamaxmbowyan ankax en: 171. ndhanracnelov xndir 170- cowyc tal, or cankaca am{ boj n ≥ 4 hamar goyowyown owni patahowyneri {A1 , A2 , . . . , An } hamaxmbowyown, orn owni het yal hatkowyownner`

a) A1 , A2 , . . . , An patahowyner xmbovin ankax en, b) A1 , A2 , . . . , An hamaxmbowyownic cankaca patahowy he{

acnelowc heto mnaca patahowyner st hamaxmbowyan k{ linen ankax: 172. A ajin sa or parownakowm 2 spitak 3 s gndikner, erkrord` 2 spitak 2 s , errord` 3 spitak 1 s gndikner: A ajin sa oric erkrord tea oxvowm patahakanoren ver{ cra mek gndik, erkrordic vercra gndik` errord, apa error{ dic` a ajin sa or: a) Oroel a ajin sa ori amenahavanakan parownakowyown, b) inpisi? havanakanowyamb bolor sa or{ neri parownakowyown kmna an o ox: 173. Masnagitakan grakanowyown oronelis, owsano oroec aycelel ereq gradaran: Yowraqanyowr gradarani hamar hava{ sarahnaravor , or ayd grakanowyown gtnvowm aynte, kam i gtnvowm, isk ee gtnvowm , apa havasarahnaravor , or zbaecra ayn ayl nercoi komic, e o: O?rn aveli havana{ kan` kgtni owsano ayd grakanowyown, e o, ee gradaranner hamalrvowm en mek myowsic ankax: 174. Apacowcel, or ee A1 ∩ A2 ∩ . . . ∩ An ⊂ A, apa P (A) ≥ P (A1 ) + P (A2 ) + . . . + P (An ) − (n − 1) :

175. Apacowcel, or cankaca erkow hamar tei owni het yal a nowyown`

|P (A ∩ B) − P (A) · P (B)| ≤

patahowyneri

176. Apacowcel, or cankaca A B patahowyneri hamar tei owni het yal anhavasarowyown` P (A ∪ B) · P (A ∩ B) ≤ P (A) · P (B) :

177. Ownenq ereq zowyg a zowyg ankax patahowyner, oronq hamate handes gal en karo: Enadrenq, or nranq bolorn l

ownen mi nowyn x havanakanowyown: Gtnel x-i me agowyn hnara{ vor areq: 178. R a avi owneco gndi nersowm patahakanoren iraric ankax nva en N keter: a) Ini? havasar kentroni nra amenamot keti mij ea

he avorowyan r -ic o pakas linelow havanakanowyown, b) Ini? havasar a) -owm stacva havanakanowyan sah{ man, erb R → ∞ RN3 → 43 πλ: 179. Molekowl, or t = 0 pahin baxvel myows molekowlin min t pah i ownecel o mi owri baxowm, λ △ t+o(△ t) havanakanow{ yamb (t, t+ △ t) amanakamijocowm kownena nor baxowm: Gtnel <azat vazqi> t-ic me linelow havanakanowyown: 180. Xowmb bakaca k tiezerakan marminneric, oroncic yowraqanyowr ankax myowsneric haytnabervowm adiolokacion kayanov p havanakanowyamb: Marminneri xowmb ditvowm ira{ ric ankax gor o m adiolokacion kayannerov: Gtnel xmbi o bolor marminneri haytnaberelow havanakanowyown: 181. Or hamakargi howsaliowyown en anvanowm hastatva

amanakamijocowm dra anxa an axatanqi havanakanowyow{ n: lektrakan an bakaca a) zowgahe , b) hajordabar miacva z1 , z2 , . . . , zk dimadrowyownneric: Yowraqanyowr dima{ drowyan howsaliowyown havasar p -i: Gtnel ayi howsaliow{ yown: 182. Sa or parownakowm a spitak, b s c karmir gndikner: Sa oric mek myowsi het ic hanowm en nra mej gtnvo bolor gndikner, nelov nranc gowyn: Gtnel spitak gndiki s ic owt er alow havanakanowyown: 183. ntrelov sa oric gndikneri dowrs hanelow hamapatas{ xan sxeman, stowgel het yal nowynowyownner`

(N −m)(N −m−1) (N −m)(N −m−1)·...·2·1 N a) 1 + NN−m + · · · + (N −1 + (N −1)(N −2) −1)(N −2)·...·(m+1)·m = m (N −m)(N −m−1) m+2 −m−1)·...·2·1 b)1+ NN−m · m+1 · m +· · ·+ (N −m)(N · m + N2 N N −m

·N m =

N m

184. Sa or parownakowm erkow gndik` spitak s : Sa{

oric hanowm en mekakan gndik min s gndiki dowrs gal, nd orowm, spitak gndiki hanelow depqowm ayn et veradarvowm avelacvowm s 2 spitak gndik: Inpisi? havanakanowyamb a ajin 50 hanva gndikner klinen spitak: 185. Sa or parownakowm n + m mianman gndik, oroncic n- spitak , isk m-` s (m ≥ n): Irar het ic a anc veradari n angam hanowm en erkowakan gndik: Gtnel amen angam tarber gowyni gndikner hanelow havanakanowyown: 186. A1 , A2 , . . . , An patahowyner ankax en st hamaxmbow{ yan P (Ak ) = pk : Inpisi?n a) A1 , A2 , . . . , An patahowyneric o meki tei ownenalow, b) A1 , A2 , . . . , An patahowyneric gone meki tei ownenalow, g ) A1 , A2 , . . . , An patahowyneric miayn meki tei ownenalow havanakanowyown: 187. Dicowq A1 , A2 , . . . , An - ankax patahowyner en P (Ai ) = pi , i = 1, 2, . . . , n: Apacowcel, or ayd patahowyneric gone meki er alow P havanakanowyown bavararowm n ∑

pi > P > 1 − e

−

n ∑

i=1

anhavasarowyownnerin: 188. In-or mek n hasceatererin namakner grel, oroncic yowraqanyowr drel a anin rari mej yowraqanyowr rari

vra patahakanoren grel n hasceneric mek: Gtnel gone mek namak it hasceov owarkelow havanakanowyown: 189. 1, 2, . . . , n ver dasavorva en patahakan kargov: In{ pisi?n veric gone meki ir teowm gtnvelow pn havanakanow{ yown: Gtnel n→∞ lim pn : 190. Patahakanoren ntrvowm n-rd kargi oroi verlow{

owyan andamneric mek: Inpisi? pn havanakanowyamb ayn i parownaki glxavor ankyownag i tarrer: Gtnel n→∞ lim pn : 191. Dahliowm ka n te: Tomser hamarakalva en bolor vaa va : Handisatesner patahakanoren zbaecnowm en te{ er: Gtnel het yal patahowyneri havanakanowyownner. a) miayn m (m ≤ n) handisates nsta klinen irenc teerowm, b) o mi handisates nsta i lini ir teowm, g) gtnel b)-owm orova havanakanowyan sahman, erb n→∞: 192.

vagonneric bakaca lektragnacq en barranowm ow or, oroncic yowraqanyowr patahakanoren nt{ rowm vagonneric mek: Gtnel yowraqanyowr vagon a nvazn mek ow or barranalow havanakanowyown: n k (k ≥ n)

Lriv havanakanowyan Bayesi bana er Dicowq ownenq (Ω, F, P ) havanakanayin tara owyown: Ee A1 , A2 , . . . , An patahowyner kazmowm en lriv xowmb P (Ai ) > 0, i = 1, ..., n, apa cankaca B patahowyi hamar tei owni lriv havanakanowyan bana ` P (B) =

n ∑

P (Ai ) · P (B/Ai ) :

i=1

Lriv havanakanowyan bana tei owni na haveli vov patahowyneri hamar` ee {Ai }∞ i=1 patahowyneri hajordaka{ nowyownn aynpisin , or

1)Ai ∩ Aj = ∅, i ̸= j, P (Ai ) > 0, ∞ ∪ Ai , 2)B ⊂ i=1

apa cankaca B patahowyi hamar P (B) =

∞ ∑

P (Ai ) · P (B/Ai ) :

i=1

Bayesi bana : Ee bavararva en lriv havanakanowyan bana i bolor paymanner P (Ak /B) =

P (B) ̸= 0,

apa

P (Ak ) · P (B/Ak ) , k = 1, 2, . . . , n : n ∑ P (Ai ) · P (B/Ai ) i=1

Bayesi bana tei owni na haveli vov {Ai }∞ i=1 pata{ howyneri hamar:

193. Dominoyi 28 qareric patahakanoren vercnowm en erkows: Inpisi? havanakanowyamb nranq kkazmen a` hamaayn xa{ i kanonneri: 194. Erkow sa or parownakowm en hamapatasxanabar m1 m2 spitak n1 n2 s gndikner: Yowraqanyowr sa oric patahakanoren vercnowm en mekakan gndik, apa ayd erkow gn{ dikneric patahakanoren ntrowm en mek: Inpisi?n ayd gndi{ ki spitak linelow havanakanowyown: 195. n gndak parownako sa ori mej gcvel mek spitak gndak: Gtnel sa oric spitak gndak hanelow havanakanowyow{ n, ee bolor hnaravor enadrowyownner spitak gndakneri skzbnakan qanaki veraberyal havasarahnaravor en: 196. Ereq sa orneric yowraqanyowr parownakowm 6 s 4 spitak gndikner: A ajin sa oric patahakanoren vercra

gndik tea oxvowm erkrord sa or, apa aydteic pata{ hakanoren vercra gndik tea oxvowm errord sa or: Gtnel errord sa oric patahakanoren hana gndiki spitak linelow havanakanowyown: 197. n sa orneric yowraqanyowr parownakowm a spitak s gndikner: A ajin sa oric patahakanoren vercra mek gndik tea oxvowm erkrord sa or, apa erkrordic` errord ayln: Verjapes verjin sa oric hanowm en mek gndik: Gtnel nra spitak linelow havanakanowyown: b

198. Owsano giti o bolor qnnakan tomser: O?r depqowm i{ maca toms vercnelow havanakanowyown klini oqragowyn, erb owsano vercnowm toms skzbowm, e? verjowm: 199. A ajin sa or parownakowm a spitak b s (a ≥ 3, b ≥ 3) gndikner, erkrord` c spitak d s gndikner: A ajin sa oric patahakanoren vercra 3 gndik tea oxowm en erkrord sa or: Gtnel erkrord sa oric patahakanoren vercra gndiki spi{ tak linelow havanakanowyown: 200. Arkowm ka enisi 15 gndak, oroncic 9- nor en: A ajin xai hamar patahakanoren vercnowm en 3 gndak, oronq xaic heto et en veradarnowm ark: Erkrord xai hamar nowynpes patahakanoren vercnowm en 3 gndak: Inpisi?n erkrord xai hamar vercra gndakneri gtagor va linelow havanakanow{ yown: 201. Aryan oxnerarkman amanak petq havi a nel do{ nori hivandi aryan xmber: IV xmbi aryown owneco hivandin kareli oxnerarkel cankaca xmbi aryown, III xmbi aryown owneco hivandin` I kam III xmbi aryown, II xmbi aryown ownecoin` I kam II , isk I xmbi aryown owneco hivandin ` miayn I xmbi aryown: Bnakowyan 33, 7%- ownen I , 37, 5%-` II , 20, 9%-` III , 7, 9%-` IV xmbi aryown:

a) Inpisi? havanakanowyamb patahakanoren ntra hi{ vandin kareli oxnerarkel patahakanoren ntra donori aryown: b) Inpisi? havanakanowyamb kareli iragor el aryan ox{ nerarkowm, ee nerka en donorneric erkows: 202. Erkow hastoc artadrowm en mianman maser, oronq owark{ vowm en ndhanowr pahest: A ajin hastoci artadroakanow{ yown erkow angam me erkrordi artadroakanowyownic: A a{ jin hastoci artadranqi 60%- barr oraki , isk erkrordini` 84%-: Pahestic patahakanoren vercra mas barr oraki : Inpisi? havanakanowyamb ayn artadrva a ajin hastoci vra: 203. Heyows artadro gor arani A, B, C hastocner artad{ rowm en amboj artadranqi hamapatasxanabar 25%, 30% 40%-: Nranc artadranqi xotan kazmowm 5%, 4% 2%: Patahakan vercra heyows xotanva : Inpisi? havanakanowyamb ayn artadrva A hastoci, B hastoci, C hastoci vra: 204. Ereq xmbaqanakneric yowraqanyowr parownakowm 10 mas: Lavorak maseri qanak a ajin, erkrord errord xmbaqa{ naknerowm havasar hamapatasxanabar 10, 7 4: Pataha{ kan xmbaqanakic patahakan vercra mas lavorak : Ayn et en veradarnowm nowyn xmbaqanakic erkrord angam vercnowm en mek mas, or nowynpes lavorak : Gtnel ayd masi errord xmba{ qanakin patkanelow havanakanowyown: 205. Hing detalneric bakaca xmbic patahakanoren verc{ nowm en mek, parzvowm or ayn xotanva : Xotanva detalne{ ri qanak havasar havanakanowyamb karo linel cankaca : Gtnel xotanva detalneri qanaki veraberyal enadrowyown{ neric amenahavanakan: 206. Haytni , or bolor tamardkanc 5%- bolor kananc gownakowyr en: Patahakanoren ntrva mard ta a{

0, 25%-

powm gownakowrowyamb: Inpisi?n havanakanowyown, or da tamard (tamardkanc kananc iv hamarel havasar): 207. entgenyan stowgowmov oqaxtavori mot oqaxt hayt{ naberelow havanakanowyown havasar 1 − β -i: A oj mar{ down hivandi te ndownelow havanakanowyown havasar α-i: Dicowq oqaxtavorner kazmowm en amboj bnakowyan γ mas: a) Gtnel mardow a oj linelow paymanakan havanakanowyown, ee stowgman amanak nran hamarel en hivand: b) Havel ha{ vanakanowyan areq masnavor depqowm, erb 1 − β = 0, 9; α = 0, 01; γ = 0, 001: 208. Masnagitacva hivandanocowm mijin havov 50%- ta{

apowm K hivandowyamb, 30%-` L hivandowyamb, 20%-` M hivandowyamb: K, L, M hivandowyownneric lriv bowvelow hava{ nakanowyownner hamapatasxanabar havasar en 0,7-i, 0,8-i, 0,9-i: Bowvoneric mek a ojaca dowrs grvowm hivandanocic: Inpisi?n havanakanowyown, or na ta apowm r K hivandow{ yamb: 209. Sa oric, or parownakowm m ≥ 3 spitak n s gndik{ ner, korel anhayt gowyni mek gndik: Orpeszi oroen sa ori parownakowyown, patahakanoren hanowm en erkow gndik: Gtnel kora gndiki spitak linelow havanakanowyown, ee haytni , or hanva gndikner spitak in: 210. Mek krakocov iraxin dipelow havanakanowyownner ereq hraigneri hamar hamapatasxanabar havasar en 54 , 34 , 3 : Hamazark talow depqowm ereq hraigneric erkows dipowm en iraxin: Gtnel errord hragi vripelow havanakanowyown: 211. Ereq hraig krakowm en, nd orowm erkows dipowm en iraxin: Gtnel errord hragi iraxin dipelow havanaka{ nowyown, ee dipelow havanakanowyownner a ajin, erkrord errord hraigneri hamar hamapatasxanabar havasar en 0,6-i, 0,5 -i 0,4-i:

212. Avtomeqena artadro gor aran owrba rerin ar{ tadrowm abaakan artadrvo avtomeqenaneri 12%-, isk erkowabiic hingabi` mnaca artadrowyown havasar qa{ nakowyamb: Artadranqi xotan owrba r kazmowm 4%, er{ kowabi hingabi rerin` 2%, isk ereqabi oreqabi rerin` 1%: Stowgman hamar vercva avtomeqenan owni gor a{ ranayin erowyown: Inpisi? havanakanowyamb ayd avtomeqe{ nan artadrvel hingabi r: 213. Dowq xndrowm eq har anin, orpeszi na jri er aikner, erb dowq gnaq hangstanalow: A anc jrelow aikner koranan 0,8 havanakanowyamb, isk jrelov oranalow havanakanowyown havasar 0,15-i: 90 tokos havanakanowyamb dowq vstah eq, or er har an i mo ana jrel aikner: Dicowq aikner oracel en: Inpisin ? havanakanowyown, or har an mo acel r jrel dranq: 214. Arkowm ka ereq metaadram: Drancic meki erkow kom <gerb> , myows metaadram kanonavor , isk errordi anrow{ yan kentron eva 75 tokosov bacvowm <gerb>-: Arkic patahakanoren ntrowm en mek metaadram netowm, ori vra bacvowm <gerb>-: Inpisi havanakanowyamb da erkow <gerb> parownako metaadramn r:

215. 1, 2, 3, . . . , n veric hajordabar patahakanoren ntrowm en erkow iv: Gtnel a ajin erkrord veri mij ea tarbe{ rowyan m-ic (m > 0) oqr linelow havanakanowyown: 216. t amanakamijocowm he axosakayanowm k kan stanalow havanakanowyown havasar Pt (k) -i: Hamarelov, or cankaca

erkow har an amanakamijocnerowm kaneri ver iraric an{ kax en, gtnel 2t amanakamijocowm s kan stanalow P2t (s) ha{ k −λt : vanakanowyown: Ini? havasar P2t (s) -, ee Pt (k) = (λt) k! e

217. Dicowq ownenq artaqin tesqov mianman n sa or, nd orowm r-rd sa or parownakowm r − 1 karmir n − r spitak gndikner: Patahakan ntrva sa oric patahakanoren ha{ jordabar a anc veradari hanowm en 2 gndik: Gtnel 2-rd han{ va gndiki spitak linelow havanakanowyown: 218. S = {1, 2, 3, . . . , N } bazmowyan bolor enabazmowyownneri hamaxmbowyownic veradarowmov ntrowyan sxemayov ntrowm en A1 A2 bazmowyownner: Inpisi? havanakanowyamb A1 ∩ A 2 = ∅: 219. Kora inqnai i oronman hamar a annacrel en 10 ow{ ai , oroncic yowraqanyowr karo gtagor vel oronman ha{ mar erkow hnaravor rjanneric mekowm, orte inqnai ka{ ro gtnvel 0,8 0,2 havanakanowyownnerov: Inpe?s petq baxel owai ner st oronman rjanneri, orpeszi inqnai i haytnaberelow havanakanowyown lini me agowyn, ee yowraqan{ yowr owai haytnaberowm oronman rjanowm gtnvo inqnai{

0,2 havanakanowyamb, isk oronowm katarvowm yowraqanyowr owai ov ankax myowsneric: Gtnel inqnai haytnaberelow havanakanowyown oronman lavagowyn tarberaki depqowm: 220. Owsanoneri xowmb bakaca a gerazancikneric, b lav c owyl sovoro owsanoneric: Qnnowyan amanak gera{ zancik stanowm miayn gerazanc gnahatakanner, lav sovoro owsano` havasar havanakanowyamb gerazanc lav gnaha{ takanner, owyl sovoro owsano` havasar havanakanowyamb lav, bavarar anbavarar gnahatakanner: Qnnowyown en han{ nowm ayd xmbic patahakanoren ntra ereq owsano: Inpisi? havanakanowyamb nranq kstanan gerazanc, lav, bavarar gnahatakanner (cankaca kargov): 221. Oro vayrowm tvyal rva eanak naxord rva pes linelow havanakanowyown havasar p-i, ee naxord r anr ayin r

q -i,

ee ayd r anr ayin r (p < 1 kam q < 1): Gtnel n-rd r anr ayin linelow pn havanakanowyown: Havel n→∞ lim pn : 222. A B xaaconeric yowraqanyowr heraxa ahelow depqowm stanowm mek miavor: A-n heraxa ahowm α hava{ nakanowyamb, B -n` β havanakanowyamb, nd orowm, α > β, α + β = 1: Amboj xa haowm ayn xaaco, or haka akordic a aj ancnowm 2 miavorov: a) Gtnel yowraqanyowr xaacoi amboj xa tanelow havanakanowyown, b) o?rn aveli ahavet A xaacoi hamar, xaal mek heraxa, e amboj xa: 223. A B xaaconer, oronq ownen hamapatasxan a b dramaglowxner, xaowm en molexa, or bakaca a anin heraxaeric: Nrancic yowraqanyowr heraxa ahowm 12 havanakanowyamb: Yowraqanyowr heraxaic heto partvo varowm haoin 1 dram: Xa arownakvowm min nrancic meki snnkacowm: Gtnel B xaacoi snnkanalow havanakanow{ yown: 224. Enadrenq naxord xndrowm A xaaco ahowm p > 12 havanakanowyamb partvowm q = 1 − p havanakanowyamb: Gtnel erkrord xaacoi snnkanalow havanakanowyown: 225. (N + 1) sa orneric yowraqanyowr parownakowm N spi{ tak s gndikner: Gndikneri baxowm st gowyni tarber bolor sa ornerowm, isk patahakan ntra sa orowm anhayt : Patahakan sa oric patahakanoren vercnowm en mek gndik tea oxowm en mek owri sa or: Ayd sa oric vercra gndik tea oxvowm mek ayl sa or aydpes arownak: Tea oxow{ yownner katarowm en (N + 1) angam aynpes, or yowraqanyowr sa or masnakcowm miayn mek angam, nd orowm (N +1)-rd tea{

oxowyown katarvowm a ajin ntra sa or: Gtnel yowra{ qanyowr sa orowm skzbnakan parownakowyan pahpanvelow ha{ vanakanowyown:

226. N hraignerin kareli baanel ors xmbi` a1 gerazan{ cik hraig, a2 lav, a3 mijak, a4 vat: Mek krakocov iraxin dipelow havanakanowyown i -rd xmbi hraigi hamar havasar pi -i, i = 1, 2, 3, 4: Erkow patahakan ntra hraigner kra{ kowm en mi nowyn iraxin: Gtnel gone mek dipowk krakoci havana{ kanowyown: 227. ors a, b, c, d mardkancic a-n stacel teekowyown, or <ayo> kam <o> azdananov haordowm b-in, b-n` c-in, c-n` d-in, isk d-n haordowm stacva teekowyown nowyn ov, inpes myowsner: Haytni , or nrancic yowraqanyowr asowm martow{ yown ereq depqeric mekowm: Oroel a ajin mardow martowyown aselow havanakanowyown, ee haytni , or orrord martow{ yown asel: 228. Anhayt gowyni n gndikner parownako sa oric hanowm en mek gndik, or spitak : Gtnel erkrord hanva gndiki spitak linelow havanakanowyown: Sa ori skzbnakan parownakowyan masin bolor enadrowyownner hamarel havasarahnaravor: 229. Anhayt gowyni n gndikner parownako sa oric hanowm en mek gndik, or spitak : Ayn et veradarnelowc heto hanowm en mek gndik s: Gtnel ayd gndiki spitak linelow havanakanowyow{ n, ee skzbnakan parownakowyan masin bolor enadrowyownne{ r havasarahnaravor en: 230. Iraric ankax axato ors tarreric bakaca sarqi tarreric erkows xa anvel en: Gtnel a ajin erkrord tarreri xa anvelow havanakanowyown, ee ayn a ajin, erkrord, errord orrord tarreri hamar hamapatasxanabar havasar ` p1 =0,1, p2 =0,2, p3 =0,3, p4 =0,4: 231. Owsowmnaranowm sovorowm en n owsano, oroncic nk -n (k = 1, 2, 3) sovorowm en k -rd tarin: Erkow patahakan ntra owsa{ noneric mek myowsic owt ndownvel: Gtnel ayd owsanoi errord tarin sovorelow havanakanowyown:

232. Kapi g ov haordvowm en AAAA, BBBB, CCCC hajor{ ∑ dakanowyownner hamapatasxanabar p1 , p2 , p3 ( pi = 1) ha{ i=1 vanakanowyownnerov: Yowraqanyowr haordvo A, B kam C ta

it ndownvowm α havanakanowyamb, isk 21 (1 − α) 21 (1 − α) havanakanowyownnerov ndownvowm myows erkow ta eri oxaren: Enadrvowm , or ta er aavavowm en iraric ankax: Inpisi? havanakanowyamb haordva eel AAAA-n, ee ndownva ABCA-n:

Be nowlii bana Ee n ankax oreric yowraqanyowrowm A patahowyi er alow havanakanowyown havasar P (A) = p (Be nowlii sxeman), isk µn - A patahowyi i hayt galow ivn n orerowm, apa P (µn = m) = Pn (m) = Cnm pm (1 − p)n−m , m = 0, 1, . . . , n :

Be nowlii bana i ndhanracowm: Dicowq katarowm en n ankax

orer, oroncic yowraqanyowrowm karo irakananal A1 , A2 , ..., As anhamateeli patahowyneric or mek: Nanakenq P (Ai ) = s ∑ pi , i = 1, ..., s, pi = 1: Havanakanowyown, or n ankax orerowm i=1 A1 patahowy kirakanana it m1 angam, A2 patahowy` it m2 angam ayln, As patahowy` it ms angam, havowm en het yal bana ov` Pn (m1 , m2 , . . . , ms ) =

n! s pm1 pm2 · · · pm s , m1 !m2 ! · · · ms ! 1 2

erb m1 + m2 + . . . + ms = n Pn (m1 , m2 , . . . , ms ) = 0, erb m1 + m2 + . . . + ms ̸= n: Ayn m-, orin hamapatasxanowm Pn (m)-i me agowyn areq, kovowm A patahowyi i hayt galow amenahavanakan iv nanakvowm mo -ov: np − (1 − p) ≤ mo ≤ np + p :

Ee p = pn = λnn , orte λn → λ (0 < λ < ∞) erb n → ∞, apa it het yal a nowyown (Powasoni eorem)` lim P (µn = m) =

n→∞

λm −λ e , m!

m = 0, 1, 2, . . . :

Mowavr-Laplasi lokal sahmanayin eorem: Ee p = const, 0 <

p < 1, q = 1 − p, apa havasaraa st ayn m-eri, oronc hamar m−np √ npq

me owyown sahmana ak , tei owni

( )) ( m − np lim P (µn = m) = √ = 1, ·φ √ n→∞ npq npq x2

orte φ(x) = √12π e− 2 fownkciayi areqner berva en xndra{ grqi ayowsak 1-owm: Mowavr-Laplasi integralayin sahmanayin eorem: Ee p = const, 0 < p < 1, q = 1 − p, apa havasaraa x1 -i x2 -i nkatmamb (∀x1 , x2 ∈ R1 ) ( ) −−−→ µn − np n → ∞ Φ(x2 ) − Φ(x1 ), P x1 < √ < x2 − npq ∫x

orte Φ(x) = √12π e− 2 du fownkciayi areqner berva en xndra{ grqi ayowsak 2-owm:

233. Za netvowm 5 angam: Inpisi?n ereqin bazmapatik vi erkow angam bacvelow havanakanowyown: 234. Katarowm en hing ankax orer, oroncic yowraqanyow{ rowm miaamanak netowm en ereq za : Ini? havasar oreric erkowsowm ereqakan miavor stanalow havanakanowyown: 235. Inpisi? havanakanowyamb 52 xaaeric bakaca

kapowk ors xaaconeri mij baanelow amanak nrancic me{ ki mot ereq angam anndmej <mekanoc> i nkni:

236. R a avi owneco rjani mej nerg va kanonavor e ankyown: Inpisi? havanakanowyamb ayd rjani mej pata{ hakanoren nva 4 keter kgtnven e ankyan mej: 237. l erkarowyown owneco AB hatva baanvowm C ketov 2:1 haraberowyamb: Ayd hatva i vra patahakanoren netowm en 4 ket: Gtnel drancic erkowsi C ketic ax, erkowsi` aj gtnvelow havanakanowyown: 238. Banaxi xndir: Mi maematikos ir mot owni lowckow erkow tow : Amen angam lowcki hanelis na patahakanoren verc{ nowm ayd tow eric mek: Oro amanak anc na nkatowm , or tow eric mek datark : Ini? havasar ayd depqowm erkrord tow i mej k hatik gtnvelow havanakanowyown, ee skzbowm yow{ raqanyowr tow i mej kar n hatik: 239. Patahowyi gone mek angam er alow havanakanowyown ors ankax orerowm havasar 0,59-i: Inpisi?n mek orowm patahowyi er alow havanakanowyown, ee yowraqanyowr or{ owm ayd havanakanowyown nowynn : 240. Katarowm en 20 ankax or, oroncic yowraqanyowrowm miaamanak netowm en 3 metaadram: Gtnel gone mek orowm ereq <gerb> bacvelow havanakanowyown: 241. Katarvowm hragowyown min a ajin dipowk krakoc: Mek krakocov iraxin dipelow havanakanowyown havasar 0,2-i: Gtnel miayn 6 krakoc talow havanakanowyown: 242. Mek krakocov iraxi <10>- xocelow havanakanowyown havasar 0,2-i: Qani? ankax krakoc petq katarel, orpeszi 0,9-ic o pakas havanakanowyamb <10>- xocvi gone mek angam: 243. Qani? angam petq netel erkow za , or gone mek angam <6, 6> bacvelow havanakanowyown lini me 12 -ic:

244. Artadranqi 5%- xotan : Oroel patahakanoren verc{ va hing artadranqneric gone erkowsi xotanva linelow hava{ nakanowyown: 245. Artadranqi 90%- lavorak , 9%- owni veracneli arat, anveracneli arat: Gtnel patahakanoren vercra ereq artadranqneric gone meki lavorak gone meki veracneli arat ownenalow havanakanowyown: 1%-`

246. Gtnel 30 aneri hamar tarva 12 amisneric 6 amisnerowm erkowakan nndyan r, isk myows 6 amisnerowm ereqakan nndyan r nknelow havanakanowyown: 247. Kanonavor metaadram netowm en 10 angam: Inpisi havanakanowyamb patahakan iragor owm kam ksksvi 3 ha{ jordakan <gerbov>, kam kavartvi erkow hajordakan <girov>: 248. Enadrenq ntaniqowm nva erexaner, ankax nta{ niqowm arden a ka myows erexaneri se ic, havasar havana{ kanowyamb karo en linel ajik kam ta: 5 erexaner owneco ntanekan zowygi hamar havel het yal patahowyneri hava{ nakanowyownner` a) bolor erexaner nowyn se i en; b) ereq me erexaner taner en, isk myowsner` ajikner; g) erexaneric it 3- taner en; d) erkow me erexaner ajikner en; e) erexaneric gone mek ajik : 249. Havanakanowyown, or hastoci vra artadrva mas klini xotanva , havasar 0,1-i: Havel patahakanoren n{ trva 10 artadranqneric amenaat meki xotanva linelow havanakanowyown (havel Be nowlii bana ov Powasoni mo{ tavor bana ov): 250. Enadrelov, or grqi mek ji vra arva tpagrakan sxal{ neri ivn owni λ = 1/2 parametrov Powasoni baxowm, havel

tvyal jowm a nvazn mek tpagrakan sxal linelow havanaka{ nowyown: 251. Hatva baanvowm maseri 1 : 2 : 3 : 4 haraberowyamb: Dra vra patahakanoren nowm en 8 ket: Gtnel a ajin hatva in 3 ket, erkrordin` 2 ket, isk mnaca keter 4-rd hatva in pat{ kanelow havanakanowyown: 252. I?nn aveli havanakan stanal` gone mek <6> ors za i netowmov, e gone mek <6, 6> erkow za eri 24 netowmnerov: 253. I?nn aveli havanakan stanal` a) gone mek <6>` za i vec netowmov, b) gone erkow <6>` za i 12 netowmov, g) gone ereq <6>` za i 18 netowmov: 254. Tvyal basketbolisti hamar gndak mek netowmov zam{ byow gcelow havanakanowyown havasar 0,4-i: Katarvel 10 netowm: Gtnel hajo oreri amenahavanakan iv nran ha{ mapatasxano havanakanowyown: 255. Yowraqanyowr orowm patahowyi er alow havanaka{ nowyown havasar 0,8-i: Oroel ankax oreri n iv, oronc depqowm patahowyi i hayt galow amenahavanakan iv hava{ sar klini 20-i: 256. Metaadram netvowm 20 angam: Gtnel <gerbi> i hayt galow amenahavanakan iv: 257. Gtnel A patahowyi 2400 ankax orerowm 1400 angam er alow havanakanowyown, ee haytni , or oreric yowraqan{ yowrowm ayd patahowyi handes galow havanakanowyown hava{ sar 0,6-i: 258. Mek krakocov iraxin dipelow havanakanowyown ha{ vasar 0,8 -i: Gtnel 100 krakocneri depqowm iraxin 75 angam dipelow havanakanowyown:

259. 200 ankax krakocneric 116- dipel en iraxin: Mek krakocov iraxin dipelow havanakanowyan o?r areqn aveli havanakan` 12 , e` 23 , ee min or katarel ayd enadrow{ yownner havasarahnaravor en miak hnaravor: 260. Za netvowm 80 angam: 0,99 havanakanowyamb gtnel ayn sahmanner, orte kgtnvi <6> -i er alow m iv: 261. Ankax oreric yowraqanyowrowm patahowyi er alow ha{ vanakanowyown havasar 0,2-i: Gtnel oreri oqragowyn n iv, ori depqowm 0,99 havanakanowyamb hnaravor lini pndel, or patahowyi er alow haraberakan haaxakanowyown kevi ir havanakanowyownic bacarak areqov o aveli, qan 0,04-ov: 262. Patahowyi er alow havanakanowyown 900 ankax or{ eric yowraqanyowrowm havasar 0,5-i: Inpisi? havanakanow{ yamb patahowyi haraberakan haaxakanowyown kevi ir er alow havanakanowyownic o avelii, qan 0,02-ov: 263. Texnikakan verahskoowyan bain stowgman enar{ kowm 475 detal: Detali xotanva linelow havanakanowyown havasar 0,05-i: 0,95 havanakanowyamb gtnel ayn sahmanner, orte kgtnvi xotanva detalneri m iv: 264. Xaaco ahowm 7 dram, ee za i vra bacvowm <6>- varowm 1 dram` haka ak depqowm: 0,999936 havanakanowyamb in? sahmannerowm kgtnvi nra aha gowmar, ee za netvowm 8000 angam:

265. Npatakaket onanowm , ee nran dipowm en yowraqan{ yowr 120 kg ki owneco erkow avia owmb, kam 200 kg ki owneco mek avia owmb: Inqnai karo be nvel o aveli, qan 1200 kg ndhanowr ki owneco cankaca miatesak avia owmberov: O?r tipi avia owmberic e ntow be nel inqnai , ee haytni , or

a ajin tipi avia owmbi dipelow havanakanowyown havasar 0,06-i, isk erkrordin` 0,08-i: 266. k hangowycneric bakaca gor iq axatel t amanak: Yowraqanyowr hangowyci howsaliowyown (anxa an axatanqi havanakanowyown) t amanakahatva owm havasar p-i: t a{ manak anc gor iq kang a nowm: Banvor stowgowm oxarinowm arqic dowrs eka hangowycner: Mek hangowyc oxarinelow hamar na axsowm τ amanak: Gtnel kang a nelowc 2τ amanak anc gor iqi axatownak linelow havanakanowyown: 267. Kapi A ket miacva 10 abonent owneco B keti het: Yowraqanyowr abonent zbaecnowm gi mijin havov amowm 6 rope: Cankaca erkow abonentneri kaner ankax en: a) Inpisi? havanakanowyamb abonentneric mek kstana merowm (gi zbava b) Gtnel anxa an spasarkman havanakanowyown, ee gi parownakowm 4 kanal: 268. Kapi A ket petq miacnel 10 abonent owneco B keti het: Yowraqanyowr abonent zbaecnowm gi ` amowm 12 rope: Cankaca erow abonentneri kanern ankax en: Gtnel anxa{

an spasarkman havanakanowyown, ee gi parownakowm 5 kanal: 269. Ereq banvor irenc hastocneri vra artadrowm en miayn gerazanc lavorak maser, nd orowm, nrancic a ajin erkror{ d artadrowm en gerazanc oraki maser` 0,9 havanakanowyamb, isk errord ` 0,8 havanakanowyamb: Banvorneric mek artad{ rel 8 mas, oroncic erkows lavorak en: Inpisi? havanakanow{ yamb nowyn banvori artadra hajord 8 maseric erkows klinen lav, isk 6-` gerazanc oraki: 270. Katarowm en 4 ankax or, oroncic yowraqanyowrowm A patahowyi handes galow havanakanowyown havasar 0,3-i:

patahowy tei ownenowm 1 havanakanowyamb, erb A pa{ tahowy handes ekel o pakas, qan 2 angam, i karo tei ownenal, erb A patahowy handes i ekel tei ownenowm 0,6 havanakanowyamb, erb A patahowy handes ekel mek angam: Gtnel B patahowyi tei ownenalow havanakanowyown: B

271. iraxi a avelagowyn miavor 10-n : Gtnel iraxin ereq krakocov 28-ic o pakas miavor stanalow havanakanow{ yown, ee 30 miavor stanalow havanakanowyown havasar 0,008-i: Haytni na , or mek krakocov 8 miavor stanalow hava{ nakanowyown havasar 0,15-i, isk 8-ic pakas miavor` 0,4-i: 272. Erkow xaaconeric yowraqanyowr netowm dram 4 an{ gam: Haowm ayn xaaco, ori mot bacva gerberi iv aveli me : Gtnel ahelow havanakanowyown yowraqanyowr xa{ acoi hamar: 273. Erkow basketbolistneric yowraqanyowr ereq angam ne{ towm gndak depi zambyow: Yowraqanyowr netman amanak gndak zambyowi mej nknelow havanakanowyown havasar ha{ mapatasxanabar 0,6-i 0,7-i: Gtnel het yal patahowyneri havanakanowyownner. a) basketbolistner katarel en havasar vov hajo ne{ towmner, b) a ajin basketbolist katarel aveli at hajo netowm{ ner, qan erkrord: 274. Erkow xankerneric yowraqanyowr n angam netowm metaadram: Gtnel nranc mot mi nowyn vov <gerb> bacvelow havanakanowyown: 275. Erkows xaowm en min haanak, nd orowm anhraet , or a ajin haanak tani m heraxaerowm, isk erkrord` n heraxaerowm: Cankaca heraxaowm ahelow havanaka{ nowyown a ajin xaacoi hamar havasar p-i, isk erkrordi

hamar` q = 1 − p: Gtnel a ajin xaacoi haanak tanelow havanakanowyown: 276. Erkow xaaco paymanavorvowm en, or ahowm kstana na, ov kahi heraxaeri oroaki qanak: Xa ndhatvel , erb a ajin xaacoin min haanak mnacel hael m, isk erkrordin` n heraxaerowm: Inpe?s baanel xaagowmar, ee cankaca heraxaowm ahelow havanakanowyown erkow xaa{ coi hamar l havasar 12 -i: 277. Yowraqanyowr orowm A patahowyi handes galow ha{ vanakanowyown havasar p-i: Gtnel n ankax orerowm A patahowyi zowyg vov handes galow havanakanowyown: 278. Mijati k ow a elow havanakanowyown havasar pk = k = 0, 1, . . ., isk vic mijat zarganalow havanakanow{ yown` p: Gtnel mijati l serownd ownenalow havanakanowyown: λk −λ , k! e

279. Mek krakocov npatakaketin dipelow havanakanowyow{ n havasar p, isk k ≥ 1 dipowmnerov npatakaket xocelow havanakanowyown` 1 − qk : Ini? havasar n krakocov npata{ kaket xocelow havanakanowyown: 280. m vnasva qneri depqowm sarq norogman kangnecnelow anhraetowyan havanakanowyown orovowm Q(m) = 1 − (1 − 1 m ω ) bana ov, orte ω -n min sarq norogman kangnecnel v{ nasva qneri mijin ivn : Apacowcel, or n artadrakan cikleric heto norogman anhraetowyan havanakanowyown orovowm Wn = 1 − (1 − ωp )n bana ov, orte p-n mek artadrakan cikli nacqowm vnasva q stanalow havanakanowyownn : 281. Sowzanav grohowm nav, arakelov hajordabar mek myowsic ankax n torped: Yowraqanyowr torped dipowm navin havanaka{ p havanakanowyamb: Torped dipelow depqowm m nowyamb jrasowzvowm navi m mekowsamaseric mek: Gtnel navi

xortakman havanakanowyown, ee dra hamar anhraet jrasowyz anel erkowsic o pakas mekowsamas: 282. Inqnai gndako vowm n ankax krakocnerov: Nrancic yowraqanyowr p1 havanakanowyamb dipowm ayn masin, orte ayn anmijapes xocowm inqnai , p2 havanakanowyamb dip{ owm va eliqi baqin p3 havanakanowyamb ndhanrapes i dipowm inqnai in: Va eliqi baqin dipa ark eq bacowm nra mej, orteic mek amowm artahosowm k litr va elanyow: Korcnelov M litr va elanyow, inqnai da nowm anmartownak: Gtnel gndako owyownic mek am anc inqnai i anmartownak linelow havanakanowyown: 283. Mrcowyan mej en mtel k hraigner, oroncic yowraqanyowr angam krakowm iraxin: Mek krakocov iraxin dipelow havanakanowyown i-rd hragi hamar havasar pi -i (i = 1, 2, . . ., k): Mrcowyown ahowm a avel at dipowmner kataro hraig: In? havanakanowyamb mrcowm khai hraigneric mek: n

284. Be nowlii sxemayowm yowraqanyowr ori hajoowyan ha{ vanakanowyown havasar p-i: Inpisi? havanakanowyamb k-rd hajoowyown tei kownena l-rd orowm: 285. Be nowlii sxemayowm hajoowyan havanakanowyown p : Gtnel 2n aydpisi orerowm m + n hajoowyownner bolor zowyg hamarner owneco orerowm ayn stanalow havanakanowyown: 286. Be nowlii sxemayowm p = 21 : Apacowcel, or √ 2 n

≤ P2n (n) <

√ 1 2n+1

287. Masniki arowm a ancqi amboj keterov ekavarvowm Be nowlii sxemayov, orte <1> elqi er alow havanakanowyown p : Masnakic ir dirqic tea oxvowm har an aj ket, ee tvyal orowm er acel <1>-, haka ak depqowm` ax ket: Gtnel

0 ketic masniki n qayleric heto m ket tea oxvelow havana{ kanowyown: 288. p havanakanowyamb yowraqanyowr vayrkyan ankax a{ manaki myows paheric anaparhov ancnowm avtomeqena: He{ tiotnin anaparh ancnelow hamar anhraet 3 vayrkyan: Inpisi? havanakanowyamb anaparhin moteco hetiotn an{ cowm katarelow hamar stipva klini spasel` a) 3 vrk., b) 4 vrk., g) 5 vrk.: 289. Be nowlii sxemayowm p-n <1> elqi havanakanowyownn , q = 1 − p-n` <0> elqi havanakanowyown: Gtnel 00 (erkow hajordakan

zro) ayi aveli owt qan 01 an irakananalow havanaka{ nowyown: Havel ayd havanakanowyown ayn masnavor depqowm, erb p = 21 : 290. 289-rd xndri paymannerowm gtnel 00 (erkow hajordakan zro) ayi aveli owt qan 10 an irakananalow havanaka{ nowyown: Havel ayd havanakanowyown p = 21 depqowm: 291. Ditarkenq ankax oreri hajordakanowyown, orte yow{ raqanyowr or za i netowmn : Inpisi? havanakanowyamb <6>-i ereq hajordakan irakanacowm tei kownena aveli owt, qan <1>-i erkow hajordakan irakanacowm: 292. Ditarkenq ankax oreri hajordakanowyown, oroncic yowraqanyowrowm or patahowy (<hajoowyown>) tei ownenowm p havanakanowyamb, isk hakadir patahowy` (<anhajoow{ yown>) q = 1 − p havanakanowyamb: I?n havanakanowyamb a hajordakan <hajoowyownner> kirakananan b hajordakan <anhajoowyownneric> aveli owt: 293. S = {1, 2, . . . , N } bazmowyownic patahakanoren mim{ yancic ankax vercvowm en erkow A1 A2 enabazmowyownner

aynpes, or S -in patkano tarr ankax myows tarreric p hava{ nakanowyamb mtnowm Ai bazmowyan mej q = 1−p havana{ kanowyamb mnowm ayd bazmowyownic dowrs: Ini? havasar A1 ∩ A2 = ∅ patahowyi havanakanowyown: 294. S = {1, 2, . . . , N } bazmowyownic enabazmowyownneri nt{ rowyan nowyn sxemayov, or berva xndir 293-owm, iraric an{ kax ntrvowm en A1 , A2 , . . . , Ar , r ≥ 2 enabazmowyownner: Gtnel ntrva enabazmowyownneri zowyg a zowyg hatvelow havana{ kanowyown: 295. Dicowq P -n P ′ - n n + 1 ankax orerowm hama{ patasxanabar A patahowyi er alow amenahavanakan vi havanakanowyownnern en (yowraqanyowr orowm P (A) = p): Apa{ cowcel, or P ′ ≤ P , nd orowm, ee (n + 1) · p-n amboj iv , apa havasarowyan nan baca vowm : 296. Kapi g ov haordowm en 100 nan: Haordman nacqowm yowraqanyowr nan karo aavavel ankax myowsneric 0,005 havanakanowyamb: Gtnel ereqic o avel nanneri aavava

linelow havanakanowyan motavor areq: 297. 130 kanal owneco kapi gi miacnowm A ket 1000 abonent owneco B keti het: Abonentneric yowraqanyowr gt{ vowm he axosic mijin amowm 6 rope: Gtnel abonentneri an{ xa an spasarkman havanakanowyown: 298. 2100 ankax oreric yowraqanyowrowm patahowyi ere{ valow havanakanowyown havasar 0,7-i: Inpisi? havanaka{ nowyamb patahowy ker a a) o pakas qan 1470 o aveli qan 1500 angam, b) o pakas qan 1470 angam, g) 1469-ic o aveli angam: 299. Ankax oreric yowraqanyowrowm patahowyi er alow ha{ vanakanowyown havasar 0,8-i: Qani? angam petq krknel

or, orpeszi 0,9 havanakanowyamb hnaravor lini pndel, or patahowy ker a o pakas, qan 75 angam: 300. iqi mek amva nacqowm asowpi het tiezeranavi baxman havanakanowyown havasar 0,001-i: Gtnel iqi ereq amsva nacqowm (hownisi 1-ic min gostosi 31-) aydpisi asowpi het baxowmneri vi vstaheli sahmanner, ee gor na{ kanapes vstahowyan havanakanowyown tvyal depqowm hava{ sar 0,995-i: 301. 1000 teanoc atron owni erkow tarber mowtqer: Yow{ raqanyowr mowtqi mot ka handeraran: Qani? te anhraet yowraqanyowr handeraranowm, orpeszi mijinowm 100-ic 99 dep{ qowm bolor handisatesner karoanan gtvel ayn mowtqi han{ derarani a ayowyownic, orteic ners en mtel: 302. Gtnel aynpisi ε > 0 iv, ori depqowm 0,99 havanaka{ nowyamb patahowyi haraberakan haaxakanowyan dra er alow havanakanowyan eman bacarak areq gerazanci ε-in: 303. Avanowm ka 2800 bnaki: Nrancic yowraqanyowr amsakan mot 6 angam gnacqov meknowm qaaq, ow orowyan rer mek myowsic ankax ntrelov patahakanoren: Gtnel gnacqi ayn oq{ ragowyn taroowyown, ori depqowm ayn ambojowyamb klcvi mijin havov 100 rowm o aveli, qan mek angam (gnacq gnowm rakan mek angam):

Patahakan me owyown baxman fownkcia Dicowq trva (Ω, F, P ) havanakanayin tara owyown: Ω-i vra orova irakan areqner ndowno a eli ξ = ξ(ω) fownk{ cian anvanowm en patahakan me owyown: Aysinqn` ξ : Ω → R1 , ξ −1 (B) = {ω : ξ(ω) ∈ B} ∈ F, ∀B ∈ B(R1 ): P {ω : ξ(ω) ∈ B} cia B -ic, B ∈ B(R1 ), kovowm

havanakanowyown, orpes fownk{ ξ patahakan me owyan hava{ nakanowyownneri baxowm kam ξ patahakan me owyan baxowm nanakvowm Pξ (B)-ov` Pξ (B) = P (ω : ξ(ω) ∈ B},

B ∈ B(R1 ) :

Masnavor depqowm, erb B = (−∞, x), P (ω : ξ(ω) ∈ (−∞, x)) = P (ω : ξ(ω) < x) havanakanowyown, orpes fownkcia x-ic, x ∈ R1 , anvanowm en ξ patahakan me owyan baxman fownkcia nanakowm en Fξ (x)-ov` Fξ (x) = P (ω : ξ(ω) < x) = P (ξ < x) :

Baxman fownkcian tva het yal hatkowyownnerov` 1. Fξ (x1 ) ≤ Fξ (x2 ), erb x1 < x2 , 2. x→−∞ lim Fξ (x) = Fξ (−∞) = 0, lim Fξ (x) = Fξ (+∞) = 1, x→+∞ 3. Fξ (x − 0) = Fξ (x): Nenq na Fξ (x)-i myows kar or hatkowyownner, oronq bxowm en 1.-3. himnakan hatkowyownneric` a) 0 ≤ Fξ (x) ≤ 1, b) P (a ≤ ξ < b) = Fξ (b) − Fξ (a) g) P (ξ = x) = Fξ (x + 0) − Fξ (x), d) Fξ (x)- karo ownenal o aveli qan haveli vov xzman keter: 1.-3. paymannerin bavararo cankaca F (x), x ∈ R1 fownk{ cia handisanowm or ξ patahakan me owyan baxman fownk{ cia` Fξ (x) = F (x):

ξ patahakan me owyownn owni diskret baxowm, ee ayn ndow{ nowm verjavor kam haveli vov areqner` x1 , x2 , ..., xk , .. hama{ patasxanabar pk = P (ξ = xk ), k = 1, 2, .., havanakanowyown{ ∑ nerov, pk = 1: Diskret ξ patahakan me owyown bnowagrvowm k

het yal ayowsakov` ξ P

x1 p1

x2 p2

xk pk

pk ≥ 0,

k = 1, 2, ...,

∑ k

pk = 1 ,

or kovowm ξ -i baxman renq: Ays depqowm Fξ (x)- astiana , nra xzman ketern en x1 , x2 , ..., xk , ..., isk yowraqanyowr xk , k = 1, 2, ..., xzman ketowm xzman me owyown havasar Fξ (xk + 0) − Fξ (xk ) = pk :

Diskret ξ patahakan me owyan baxman fownkcian owni he{ t yal tesq` ∑ Fξ (x) =

pk :

k:xk <x

patahakan me owyownn owni bacarak anndhat baxowm, ee goyowyown owni o bacasakan borelyan fξ (x) fownkcia ayn{ pisin, or cankaca borelyan B ∈ B(R1 ) bazmowyan hamar ξ

∫

Pξ (B) = P (ξ ∈ B) =

fξ (x)dx, B

∫

orte fξ (x) dx = 1: fξ (x)- anvanowm en ξ patahakan me ow{ R1 yan xtowyan fownkcia kam ξ patahakan me owyan xtowyown: Masnavor depqowm, erb B = (−∞, x)` ∫x Fξ (x) =

fξ (u)du, −∞

orte

∫∞ −∞

fξ (x) dx = 1:

Cankaca f (x), x ∈ R1 , fownkcia, orn tva het yal hatkowyownnerov` 1. f (x) ≥ 0, ∫∞ 2. fξ (x) dx = 1 −∞ handisanowm or ξ patahakan me owyan xtowyown` fξ (x) = f (x): Berenq mi qani haytni baxowmneri rinakner:

Diskret baxowmner 1. Binomakan baxowm p, n parametrerov (n- bnakan iv , 0 ≤ p ≤ 1)` P (ξ = k) = Cnk pk (1 − p)n−k ,

k = 0, 1, 2, ..., n :

2. Powasoni baxowm λ > 0 parametrov (ξ ∼ Π(λ))` P (ξ = k) =

λk −λ e , k!

k = 0, 1, 2, ...

3. Erkraa akan baxowm p parametrov (0 ≤ p ≤ 1)` P (ξ = k) = (1 − p)k−1 · p,

k = 1, 2, ...

4. Hipererkraa akan baxowm n, M, N parametrerov (n-, M -, N - bnakan ver en)` P (ξ = k) =

k C n−k CM N −M , n CN

k = max(0, n − N + M ), ..., min(M, n) :

Bacarak anndhat baxowmner 1. Havasaraa baxowm [a, b] mijakayqowm`   0, F (x) =

x−a ,  b−a



x≤a a<x≤b, x>b

{ f (x) =

b−a ,

x ∈ (a, b) x∈ / (a, b)

2. Cowcayin baxowm λ > 0 parametrov` {

F (x) =

1 − e−λx , 0,

x≥0 , x<0

{

f (x) =

λe−λx , 0,

x≥0 x<0:

3. Normal (Gaowsi) baxowm (a, σ) parametrerov` −∞ < a < +∞, σ 2 > 0, (ξ ∼ N (a, σ 2 ))` { } (x − a)2 f (x) = √ exp − , 2σ 2 σ 2π

−∞ < x < +∞ :

ξ ∼ N (0, 1) patahakan me owyownn anvanowm en standart normal

patahakan me owyown: 4. Koii baxowm (a, σ) parametrerov` −∞ < a < +∞, σ > 0 f (x) =

σ , · π σ 2 + (x − a)2

−∞ < x < ∞ :

Standart Koii baxowm` f (x) =

, π(1 + x2 )

−∞ < x < ∞ :

5. Gamma baxowm (α, λ) parametrerov` α > 0, λ > 0, (ξ ∼ Γ(α, λ))` { f (x) =

orte Γ(λ) = veri hamar

αλ λ−1 −αx e , Γ(λ) x

x≥0

x < 0,

∫∞

xλ−1 e−x dx- yleri gamma fownkcian , n bnakan √ Γ(n) = (n − 1)! Γ( 12 ) = π :

304. Metaadram netvowm erkow angam: Ka owcel gerbi handes galow vi baxman renq: Gtnel baxman fownkcian ka owcel dra grafik:

305. Dicowq 10 manrakneric 8- standart en: Patahakanoren vercnowm en erkow manrak: Kazmel vercva manrakneri mej stan{ dart manrakneri vi baxman renq: Gtnel baxman fownkcian ka owcel dra grafik: 306. Artadranqi 10%- xotan : Patahakanoren vercnowm en ors manrak: Gtnel dranc mej xotan manrakneri vi bax{ man renq: 307. Hing sarqeri howsaliowyown stowgelow hamar katarowm en hajordakan ankax orarkowmner: Yowraqanyowr hajord sar{ q stowgman enarkvowm miayn ayn depqowm, erb naxord howsali : Kazmel stowgva sarqeri vi baxman renq, ee orarkow{ m ancnelow havanakanowyown drancic yowraqanyowri hamar havasar 0,9-i: 308. Metaadram netowm en min gerbi a ajin angam ere{ val: Dicowq ξ -n netowmneri ivn : Kazmel ξ patahakan me ow{ yan baxman renq, gtnel P (ξ > 1) havanakanowyown: 309. Hragi mek krakocov iraxin dipelow havanakanow{ yown havasar 0,8-i: Hraig krakowm min a ajin vripowm: Kazmel krakocneri vi baxman renq: 310. Erkow basketbolist hajordabar netowm en gndak depi zambyow min a ajin hajoowyown: Ka owcel netowmneri vi baxman renq yowraqanyowr basketbolisti hamar, ee hajo{ owyan havanakanowyown a ajin basketbolisti hamar hava{ sar 0,4-i, isk erkrordi hamar` 0,6-i: or

311. Dicowq F (x)- anndhat baxman fownkcia : Apacowcel, ∫∞ F (x)dF (x) = −∞

312. Apacowcel, or cankaca anndhat F (x) baxman fownk{ ciayi hamar cankaca bnakan n k veri hamar tei owni het yal a nowyown` ∫∞ F k (x)dF n (x) = −∞

n : n+k

313. ξ patahakan me owyan baxman fownkcian havasar   0, F (x) = x2 ,   1,

x ≤ 0, 0 < x ≤ 1, x>1:

Gtnel ors ankax oreric ereqowm ξ patahakan me owyan ndowna areqneri [0, 25; 0, 75] mijakayqowm gtnvelow havana{ kanowyown: 314. Trva in-or patahakan me owyan baxman fownk{ cian 

F (x) =

 0,      1/2,    3/5,

 4/5,      9/10,    1,

x≤0 0<x≤1 1<x≤2 2<x≤3 3 < x ≤ 3, 5 x > 3, 5 :

Gtnel ayd patahakan me owyan baxman ayowsak: 315. Dicowq trva ξ patahakan me owyan xtowyan fownk{ cian { f (x) =

c(1 − x2 ), 0,

−1 < x < 1

mnaca depqerowm:

a) Gtnel c gor akic: b) Gtnel F (x) baxman fownkcian: 316. Hnarav?or ardyoq ntrel c hastatownn aynpes, or cx−3 fownkcian irenic nerkayacni havanakanowyownneri baxman xtowyown het yal bazmowyownneri vra` a) [1, ∞) a agayi b) [0, ∞) a agayi, g) [−2, −1] hatva i vra: 317. ξ patahakan me owyan havanakanowyownneri baxman xtowyownn ` f (x) =

a : 1 + x2

Gtnel a) a gor akic, b) F (x) baxman fownkcian, g) ξ < 1) havanakanowyown:

P (−1 ≤

318. ξ patahakan me owyan havanakanowyownneri baxman xtowyownn ` f (x) =

: π(1 + x2 )

Havel a) P (ξ ≥ 1), b) P (|ξ| ≥ 1): 319. ξ patahakan me owyan havanakanowyownneri baxman xtowyownn ` f (x) =

e−x

a : + ex

Gtnel a gor akic erkow ankax orerowm ξ patahakan me{

owyan mekic oqr areq ndownelow havanakanowyown: 320. ξ patahakan me owyown baxva <owankyown e an{ kyan renqov> [0, a] mijakayqowm` {

f (x) =

0, ) ( c 1 − xa ,

x∈ / (0, a) x ∈ (0, a) :

Gtnel a) c hastatown, b) F (x) baxman fownkcian, g) P (a/2 ≤ havanakanowyown:

ξ ≤ a)

321. ξ patahakan me owyown baxva Simpsoni renqov` <havasarasrown e ankyan renqov> [−a, a] mijakayqowm`  ( ) x  c (1 + a ) , f (x) = c 1 − xa ,   0,

x ∈ (−a, 0) x ∈ (0, a) x∈ / (−a, a),

Gtnel a) c hastatown, b) F (x) baxman fownkcian, g) P (a/2 ≤ havanakanowyown:

ξ < a)

322. ξ patahakan me owyownn ndownowm o bacasakan amboj areqner: Apacowcel het yal pndowmneri hamareqowyown` a) ξ -n owni havanakanowyownneri erkraa akan baxowm, b) P (ξ = n + k/ξ ≥ k) = P (ξ = n), k = 0, 1, 2, ..., n = 0, 1, 2, ...: 323. ξ patahakan me owyownn owni λ = 1/3 parametrov havanakanowyownneri cowcayin baxowm: Gtnel a) P (ξ > 3), b) P (ξ > 6/ξ > 3) g) P (ξ > t + 3/ξ > t), orte t > 0 irakan iv : 324. Dicowq ξ -n cowcayin baxowm owneco patahakan me ow{ yown , isk t > 0 irakan iv : Gtnel (ξ − t)-i baxman fownkcian ξ ≥ t paymani depqowm: 325. Diskret ξ patahakan me owyownn owni het yal baxman renq` ξ -2 -1 0 1 2 P 0,1 0,2 0,3 0,3 0,1 Gtnel a) η1 = ξ 2 + 1, b) η2 = |ξ| patahakan me owyownneri baxman renqner: 326. ξ patahakan me owyan F (x) baxman fownkcian an{ ndhat 0 ketowm: Inp?es baxva

{

η=

ξ |ξ| ,

ee ee

ξ ̸= 0 ξ=0

patahakan me owyown: 327. ξ patahakan me owyown havasaraa baxva [0, 1] mijakayqowm: Gtnel a) η = ξ 2 , b) η = 1/ξ , g) η = eξ patahakan me owyownneri havanakanowyownneri baxman xtowyownner: 328. ξ patahakan me owyown havasaraa baxva [0, 1] ξ ), g) η = − λ1 ln(1− ξ) mijakayqowm: Gtnel a) η = ln ξ −1 , b) η = ln( 1−ξ (orte λ > 0) patahakan me owyownneri havanakanowyownneri baxman xtowyownner: 329. ξ patahakan me owyown havasaraa baxva [−π/2, mijakayqowm: Gtnel a) η = sin ξ , b) η = | sin ξ| patahakan me owyownneri havanakanowyownneri baxman xtowyownner:

π/2]

330. ξ patahakan me owyown havasaraa baxva [−1, 1] mijakayqowm: Gtnel η = |ξ| patahakan me owyan baxman fownk{ cian: 331. ξ patahakan me owyown havasaraa baxva [0, 2] mijakayqowm: Gtnel η = |ξ − 1| patahakan me owyan baxman fownkcian: 332. ξ patahakan me owyownn owni standart Koii baxowm` f (x) =

, π(1 + x2 )

xtowyan fownkciayov: Gtnel xtowyown:

(−∞ < x < +∞)

η = 1/ξ

patahakan me owyan

333. l erkarowyown owneco o patahakan ketov baanva

erkow masi: Gtnel ayn owankyan makeresi baxman fownkcian, ori hamar komer en handisanowm oic stacva maser: 334. [0, a] mijakayqi vra patahakanoren nowm en erkow ket` aysinqn, ayd keteri abscisner havasaraa en baxva [0, a]

mijakayqowm: Gtnel ayd keteri mij ea he avorowyan baxman fownkcian havanakanowyownneri baxman xtowyown: 335. [0, a] mijakayqi vra patahakanoren nowm en n ket: Gtnel axic k-rd keti abscisi havanakanowyownneri baxman xtowyown: 336. (0, a) ketic OY a ancqin φ ankyan tak tarva owi gi : Gtnel ayd owi OX a ancqi het hatman keti abscisi baxman fownkcian baxman xtowyown, ee φ ankyown hava{ saraa baxva a) [0, π/2] mijakayqowm, b) [−π/2, π/2] mija{ kayqowm: 337. rdinatneri a ancqi (0, 0) (0, R) keteri mij pata{ hakanoren nva mi ket: Ayd ketov OY a ancqin owahayac tarva x2 + y2 = R2 rjanag i lar: Gtnel lari erkarowyan havanakanowyownneri baxman xtowyown: 338. R a avov (0, 0) kentronov rjanag i vra pataha{ kanoren nowm en mi ket, ayd keti b e ayin ankyown havasara{ a baxva [−π, π]-owm: Gtnel nva keti abscisi havana{ kanowyownneri baxman xtowyown: 339. Dicowq ξ patahakan me owyan baxman fownkcian ha{ vasar `   0,      x/4, F (x) = 12 + x−1 4 ,   11/12,     1,

Gtnel P (ξ = i), i = 1, 2, 3

x≤0 0<x≤1 1<x≤2 2<x≤3 x > 3.

P (1/2 ≤ ξ < 3/2):

340. Avtobowsner, oronc axatanqayin am sksvowm a a{ votyan am 7-in, kanga in en motenowm 15 rope ndmijowmov:

Aysinqn nranq arvowm en am 7-in, 7:15, 7:30, 7:45 ayln: Enad{ relov kanga in motenalow pahi havasaraa baxva owyown am 7-ic 7:30 mijakayqowm, havel havanakanowyown, or na kspasi a) o aveli qan 5 rope, b) 10 ropeic aveli: 341. Dicowq ξ patahakan me owyown havasaraa bax{ va (0, 10) mijakayqowm: Havel ξ < 3, ξ > 6, 3 < ξ < 8 patahowy{ neri havanakanowyown: 342. Dicowq ξ -n a = 3, σ2 = 9 parametrerov normal baxva

patahakan me owyown : Havel P (|ξ − 3| > 6): 343. Dicowq ξ -n a = 10, σ2 = 36 parametrerov normal baxva

patahakan me owyown : Havel a) P (ξ > 5); b) P (4 < ξ < 16); g) P (ξ < 8); d) P (ξ < 20); e) P (ξ > 16):

B 344. Dicowq ξ -n η-n mi nowyn (Ω, F, P ) havanakanayin ta{ ra owyan vra orova patahakan me owyownner en: Apacowcel, or A = {ω : ξ(ω) < η(ω)}-n, B = {ω : ξ(ω) = η(ω)}-n, C = {ω : ξ(ω) ≤ η(ω)}-n patahowyner en: 345. ξ η patahakan me owyownner orova en mi nowyn (Ω, F, P ) havanakanayin tara owyan vra: Apacowcel, or a) ξ + η, b) ξ − η, g) ξ · η, d) max(ξ, η), e) min(ξ, η) patahakan me owyownner en:

346. Dicowq ξ1 , ξ2 , ..., ξn , ... patahakan me owyownner orova

en mi nowyn (Ω, F, P ) havanakanayin tara owyan vra: Apacow{ cel, or inf ξn , n

sup ξn , n

patahakan me owyownner en:

lim ξn , n→∞

lim ξn

n→∞

347. Dicowq ξ -n patahakan me owyown , isk f (x)- borelyan fownkcia : Apacowcel, or η = f (ξ)-n patahakan me owyown : 348. Katarva

oreri patahakan me owyown ` P (ξ = k) =

iv Powasoni baxowm owneco

λk −λ e , k!

k = 0, 1, 2, ... :

Yowraqanyowr or p havanakanowyamb karo linel hajo (1−p) havanakanowyamb` anhajo: Ka owcel hajo oreri vi baxman renq: 349. Dicowq F (x)- baxman fownkcia : Apacowcel, or G1 (x) = h

∫

x+h

F (u)du, x

G2 (x) = 2h

x+h ∫

F (u)du x−h

fownkcianer cankaca h > 0 depqowm handisanowm en baxman fownkcianer: 350. Dicowq Apacowcel, or

F (x)-

baxman fownkcia , nd orowm {

G(x) =

F (x) − F ( x1 ), 0,

F (0) = 0:

x≥1 x<1

fownkcian handisanowm baxman fownkcia: 351. Apacowcel, or ee baxman fownkcian anndhat owi yowraqanyowr ketowm, apa ayn havasaraa anndhat amboj owi vra: 352. Apacowcel, or cankaca baxman fownkcia karo owne{ nal o aveli, qan haveli vov xzman keter:

353. Kar?o ardyoq baxman fownkciayi xzman keteri baz{ mowyown linel amenowreq xit owi vra: 354. Dicowq F (x)- ξ patahakan me owyan baxman fownk{ cian : Gtnel 12 (ξ + |ξ|) patahakan me owyan baxman fownk{ cian: 355. Dicowq F (x)- ξ patahakan me owyan baxman fownk{ cian : Gtnel η = aξ + b patahakan me owyan baxman fownk{ cian: 356. Dicowq ξ patahakan me owyan F (x) baxman fownkcian anndhat : Gtnel η = F (ξ) patahakan me owyan baxman fownkcian: 357. rjani a avi a va motavorapes: Gtnel rjani makeresi baxman fownkcian, hamarelov a avi havasara{ a baxva [a, b] mijakayqowm: 358. Apacowcel, or normal baxowm owneco patahakan me{

owyownic g ayin fownkcian s normal : 359. ξ patahakan me owyownn owni normal baxowm (a, σ2 ) pa{ rametrerov: Gtnel signξ patahakan me owyan baxman fownk{ cian: 360. Dicowq ξ ∼ N (0, 1) standart normal patahakan me ow{ yown : Gtnel η = ξ 2 patahakan me owyan havanakanowyownneri baxman xtowyown: 361. ξ patahakan me owyownn owni normal baxowm (a, σ2 ) pa{ rametrerov: Gtnel a) η = ξ 2 , b) η = ξ 3 patahakan me ow{ yownneri havanakanowyownneri baxman xtowyownner: 362. ξ patahakan me owyownn owni cowcayin baxowm λ = 1 parametrov: Gtnel η = e−ξ patahakan me owyan havanaka{ nowyownneri baxman xtowyown:

363. Dicowq ξ patahakan me owyan havanakanowyownneri f (x) baxman xtowyownn ` {

f (x) =

0, c 2 (x + 1),

x∈ / (−1, 1) x ∈ (−1, 1) :

Gtnel c hastatown η = 1 − ξ 2 patahakan me owyan hava{ nakanowyownneri baxman xtowyown:

Bazmaa patahakan me owyown baxman fownkcia Mi nowyn (Ω, F, P ) havanakanayin tara owyan vra orova

ξ1 (ω), ξ2 (ω), . . . ,)ξn (ω) patahakan me owyownneri ξ(ω) = (ξ1 (ω), ξ2 (ω), . . ., ξn (ω) hamaxmbowyown kovowm n a ani pataha{ kan me owyown kam patahakan vektor: ( ) Pξ (B) = P (ω : ξ(ω) ∈ B) = P (ω : ξ1 (ω), ξ2 (ω), . . . , ξn (ω) ∈ B) − n

B ∈ B(Rn ),

anvanowm en ξ = (ξ1 , ξ2 , . . . , ξn ) patahakan vektori

baxowm: Masnavor depqowm, erb B = (−∞, x1 ) × (−∞, x2 ) × . . . × (−∞, xn ) Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) = P (ω : ξ1 (ω) < x1 , ξ2 (ω) < x2 , . . . ,

ξn (ω) < xn ) = P (ξ1 < x1 , ξ2 < x2 , . . . , ξn < xn )- xk ∈ R, k = 1, 2, . . . , n, anvanowm en ξ = (ξ1 , ξ2 , . . . , ξn ) patahakan vek{ tori baxman fownkcia kam ξ1 , ξ2 , . . . , ξn patahakan me owyown{

neri hamate baxman fownkcia: Bazmaa baxman fownkcian tva het yal hatkow{ yownnerov` 1. Fξ1 ,ξ2 ,...,ξk−1 ,ξk ,ξk+1 ,...,ξn (x1 , x2 , . . . , xk−1 , xk , xk+1 , . . . , xn ) ≤ ≤ Fξ1 ,ξ2 ,...,ξk−1 ,ξk ,ξk+1 ,...,ξn (x1 , x2 , . . . , xk−1 , x′k , xk+1 , . . . , xn ), erb xk < x′k , k = 1, 2, . . . , n : 2. lim Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) = 0, erb xi -ic gone mek gtowm −∞ lim

x1 →+∞ x2 →+∞ xn →+∞

Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) = 1 :

3. Fξ1 ,ξ2 ,...,ξk−1 ,ξk ,ξk+1 ,...,ξn (x1 , x2 , . . . , xk−1 , xk − 0, xk+1 , . . . , xn ) = = Fξ1 ,ξ2 ,...,ξk−1 ,ξk ,ξk+1 ,...,ξn (x1 , x2 , . . . , xk−1 , xk , xk+1 , . . . , xn ), k = 1, 2, . . . , n : 4. Cankaca a = (a1 , a2 , . . . , an )-i b = (b1 , b2 , . . . , bn )-i ak ≤ bk , k = 1, 2, . . . , n

hamar`

△a1 b1 . . . △an bn Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) ≥ 0,

orte

△ak bk Fξ1 ,ξ2 ,...,ξk−1 ,ξk ,ξk+1 ,...,ξn (x1 , x2 , . . . , xk−1 , xk , xk+1 , . . . , xn ) = = Fξ1 ,ξ2 ,...,ξk−1 ,ξk ,ξk+1 ,...,ξn (x1 , x2 , . . . , xk−1 , bk , xk+1 , . . . , xn ) − − Fξ1 ;ξ2 ;...;ξk−1 ;ξk ;ξk+1 ;...;ξn (x1 , x2 , . . . , xk−1 , ak , xk+1 , . . . , xn )

kovowm tarberakan perator: Nenq na Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) fownkciayi myows kar or hatkowyownner` a) Fξ1 ,ξ2 ,...,ξi ,...,ξj ,...,ξn (x1 , x2 , . . . , xi , . . . , xj , . . . , xn ) = = Fξ1 ,ξ2 ,...,ξj ,...,ξi ,...,ξn (x1 , x2 , . . . , xj , . . . , xi , . . . , xn ) (hamaa owyan hatkowyown) b) Fξ1 ,...,ξn−1 ,ξn (x1 , x2 , . . . , xn−1 , +∞) = Fξ1 ,ξ2 ,...,ξn−1 (x1 , x2 , . . . , xn−1 ) (hamaaynecva owyan hatkowyown): 1 - 4 paymannerin bavararo cankaca F (x1 , x2 , . . . , xn ) fownkcia, xk ∈ R, k = 1, 2, . . . , n handisanowm or ξ = (ξ1 , ξ2 , . . ., ξn ) patahakan vektori baxman fownkcia` Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) = F (x1 , x2 , . . . , xn ) : ξ = (ξ1 , ξ2 , . . . , ξn ) patahakan vektor diskret , ee yowra{ qanyowr ξk -n, k = 1, 2, . . . , n diskret patahakan me owyown : Ee ξ = (ξ1 , ξ2 , . . . , ξn )-n diskret , apa cankaca B ∈ B(Rn )

hamar

Pξ (B) = P (ξ ∈ B) =

∑

pi1 ,i2 ,...,in ,

i1 ,i2 ,...,in : (ki ,ki ,...,kin )∈B

orte pi1 ,i2 ,...,in = P (ξ1 = ki1 , ξ2 = ki2 , . . . , ξn = kin ),

isk kij -n ξj , j = 1, ..., n patahakan me owyan hnaravor areq{ neric mekn : ξ = (ξ1 , ξ2 , . . . , ξn ) patahakan vektor bacarak anndhat , ee goyowyown owni o bacasakan borelyan fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) fownkcia aynpisin, or cankaca B ∈ B(Rn ) hamar ∫ Pξ (B) = P (ξ ∈ B) =

fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) dx1 dx2 . . . dxn , B

ayste ∫ fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) dx1 dx2 . . . dxn = 1 : Rn

Masnavor depqowm, erb B = (−∞, x1 ) × (−∞, x2 ) × . . . × (−∞, xn ) Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) = x ∫x1 ∫n fξ1 ,ξ2 ,...,ξn (u1 , u2 , . . . , un )du1 du2 . . . dun ,

−∞

−∞ ∫∞

ayste ξ2 ,

−∞

∫∞ −∞

fξ1 ,ξ2 ,...,ξn (u1 , u2 , . . . , un )du1 du2 . . . dun = 1:

fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) fownkcian kovowm n-a ani ξ = (ξ1 , . . . , ξn ) patahakan me owyan xtowyown: Cankaca f (x1 , x2 , . . . , xn ), xk ∈ R, k = 1, 2, . . . , n fownkcia,

or bavararowm het yal hatkowyownnerin` 1.f (x1 , x2 , . . . , xn ) ≥ 0, ∫∞ ∫∞ 2. . . . f (x1 , x2 , . . . , xn ) dx1 dx2 . . . dxn = 1 −∞ −∞ handisanowm or patahakan vektori xtowyown: Nenq, or fξk (xk ) =

∫∞

−∞

∫∞

−∞

fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) dx1 . . . dxk−1 dxk+1 dxn :

Berenq haytni bazmaa baxman rinak. Erka normal (Gaowsi) baxowm a1 , a2 , σ1 , σ2 , r (σ1 > 0, σ2 > 0, |r| < 1) parametrerov`

fξ1 ,ξ2 (x1 , x2 ) = (x2 −a2 )2 ) ] : σ2

1√

2πσ1 σ2

1−r2

( (x1 −a1 )2 2r(x1 −a1 )(x2 −a2 ) exp − 2(1−r − + 2) [ σ1 σ2 σ2

Patahakan me owyownneri ankaxowyown ξ1 , ξ2 , . . . , ξn patahakan me owyownner kovowm en ankax, ee B1 ∈ B(R) ,B2 ∈ B(R), . . . , Bn ∈ B(R) hamar P (ξ1 ∈ B1 , . . . , ξn ∈ Bn ) = P (ξ1 ∈ B1 )P (ξ2 ∈ B2 ) · · · P (ξn ∈ Bn ) :

patahakan me owyownnern ankax en ayn ayn depqowm, erb

ξ1 , ξ2 , . . . , ξn

miayn

Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) = Fξ1 (x1 )Fξ2 (x2 ) · · · Fξn (xn ) :

Diskret patahakan me owyownneri ankaxowyan hamar an{ hraet bavarar, orpeszi P (ξ1 = ki1 , . . . , ξn = kin ) = P (ξ1 = ki1 )P (ξ2 = ki2 ) · · · P (ξn = kin ) :

Bacarak anndhat patahakan me owyownneri ankaxowyan hamar anhraet bavarar, orpeszi fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) = fξ1 (x1 )fξ2 (x2 ) · · · fξn (xn ), xk ∈ R,

k = 1, 2, . . . , n :

Paymanakan baxowmner Diskret depq. Ee ξ -n η-n diskret patahakan me owyownner en, apa ξ patahakan me owyan baxowm η = y paymani depqowm sahmanvowm het yal bana ov` pξ/η (x, y) = P (ξ = x/η = y) =

p(x, y) P (ξ = x, η = y) = , P (η = y) pη (y)

y -i

bolor ayn areqneri hamar, orte pη (y) > 0: ξ patahakan me owyan paymanakan baxman fownkcian η = y paymani dep{ qowm sahmanvowm het yal bana ov` Fξ/η (x, y) = P (ξ < x/η = y) =

∑

pξ/η (a/y)

a<x

y -i

bolor ayn areqneri hamar, orte pη (y) > 0: ξ η patahakan me owyownnern ownen hamate f (x, y) xtowyan fownkcia: ξ patahakan me ow{ yan paymanakan xtowyown η = y paymani depqowm sahmanvowm het yal bana ov`

Anndhat depq. Dicowq

fξ/η (x/y) =

f (x, y) , fη (y)

y -i bolor ayn areqneri hamar, orte fη (y) > 0: Masnavorapes, ∫ a Fξ/η (a, y) = fξ/η (x/y) dx : −∞

364. sar

(ξ, η)

patahakan vektori baxman xtowyown hava{ f (x, y) =

Gtnel c-n

F (x, y)

c (16 +

x2 )(25

+ y2)

baxman fownkcian:

365. (ξ, η) patahakan vektori baxman fownkcian havasar { F (x, y) =

1 − 2−x − 2−y + 2−x−y , 0,

x ≥ 0, y ≥ 0

mnaca depqerowm:

1) Gtnel (ξ, η) patahakan keti x = 1, x = 2, y = 3, y = 5 owinerov sahmana akva owankyan patkanelow havana{ kanowyown:

2) Gtnel erka baxman f (x, y) xtowyown: 3) Gtnel (ξ, η) patahakan keti A(1; 3), B(3; 3), C(2; 8) gaga{ nerov e ankyan patkanelow havanakanowyown: 366. (ξ, η) patahakan vektori havanakanowyownneri bax{ man xtowyown havasar f (x, y) =

a : 1 + x2 + y 2 + x2 y 2

Gtnel a gor akic: Gtnel ξ η patahakan me owyownneri mek a ani baxman xtowyownner: Anka?x en ardyoq ξ -n η-n: 367. (ξ, η) patahakan vektori havanakanowyownneri bax{ man xtowyown havasar { xe−x(1+y) , f (x, y) = 0,

x > 0, y > 0

mnaca depqerowm:

Havel a) ξ patahakan me owyan havanakanowyownneri bax{ man xtowyown, b) η patahakan me owyan havanakanowyown{ neri baxman xtowyown: 368. (ξ, η) patahakan vektori havanakanowyownneri bax{ man xtowyown havasar {

f (x, y) =

Apacowcel, or ξ -n

24xy(1 − x)2 , 0, η -n

0 ≤ x ≤ 1, 0 ≤ y ≤ 1

mnaca depqerowm:

ankax en:

369. (ξ, η) patahakan vektori havanakanowyownneri bax{ man xtowyown havasar {

f (x, y) =

24x2 y(1 − x), 0,

0 < x < 1, 0 < y < 1

mnaca depqerowm:

Apacowcel, or ξ -n

η -n

ankax en:

370. Netowm en 2 kanonavor za : Gtnel ξ η patahakan me owyownneri hamate havanakanowyownneri baxowm, ee` a) ξ -n erkow za eri vra bacva miavorneric me agowynn , isk η-n erkow za eri vra bacva miavorneri gowmarn : b) ξ -n a ajin za i vra bacva miavorn , isk η-n erkow za eri vra bacva miavorneric me agowynn : 371. Sa oric, or parownakowm 5 spitak 8 karmir gndik{ ner, a anc veradarman hanowm en 2 gndik: Dicowq ξi -n ndownowm 1 areq, ee ntrva i-rd gndik spitak , 0 areq` haka{

ak depqowm: Gtnel ξ1 ξ2 -i hamate havanakanowyan baxow{ m: 372. Dicowq ξ , η ζ -n ankax patahakan me owyownner en, nd orowm ζ -n ndownowm 1 kam 0 areqner hamapatasxanabar p q havanakanowyownnerov, p + q = 1, ξ -n η -n ownen hamapa{ tasxanabar F (x) G(x) baxman fownkcianer: Gtnel het yal patahakan me owyownneri baxman fownkcianer` a) ζξ + (1 − ζ)η, b) ζξ + (1 − ζ) max{ξ, η}, g) ζξ + (1 − ζ) min{ξ, η} :

373. Apacowcel, or patahakan me owyownn ankax inqn ire{ nic ayn miayn ayn depqowm, erb ayn 1 havanakanowyamb has{ tatown : 374. Dicowq ξ -n me owyownner en

η -n miatesak baxva ankax patahakan P (ξ > 0) = P (η > 0) ≥ 1/2: it ? ardyoq, or

P (ξ + η > 0) ≥ 1/2:

375. Dicowq ξ -n η-n patahakan me owyownner en, nd orowm P (ξ > 0) = P (η > 0) = 3/4 P (ξ + η > 0) = 1/2: Apacowcel, or ξ -n η -n ankax en:

376. Dicowq ξ η patahakan me owyownner 1 havanaka{ nowyamb havasar en hastatownneri, nd orowm P (ξ < η) = 1: Karo? en ξ -n η-n linel ankax: ξ -n

377. Dicowq ξ , η, ζ patahakan me owyownnern aynpisin en, or ankax (η + ζ)-ic: Ankax ? ardyoq ξ -n η-ic ζ -ic:

378. Dicowq ξ , η, ζ patahakan me owyownnern aynpisin en, or ξ -n ankax η -ic ζ -ic: Ankax ? ardyoq ξ -n (η + ζ)-ic: 379. Goyowyown owne?n ardyoq aynpisi ξ η patahakan me{

owyownner, or 1 havanakanowyamb ξ -n η-n hastatownner en a) ξ -n (ξ + η)-n ankax en, b) ξ -n (ξ · η)-n ankax en, g) ξ -n, (ξ + η)-n (ξ · η)-n st hamaxmbowyan ankax en: 380. Dicowq trva het yal havanakanayin tara owyown`

Ω = {1, 2, 3, 4}, F - Ω-i bolor enabazmowyownneri dasn , P (1) = P (2) = P (3) = P (4) = 1/4: Ka owcel ayd tara owyan vra erkow

ankax patahakan me owyownner, oronq 1 havanakanowyamb havasar linen hastatownneri: 381. Dicowq ξ -n η-n ankax patahakan me owyownner en, f (x)- g(x)- borelyan fownkcianer en: Apacowcel, or f (ξ)-n g(ξ)-n nowynpes ankax en: 382. Dicowq ξ, η, ζ patahakan me owyownnern st hamaxmbow{ yan ankax en: Ankax kline?n ardyoq ξ (η + ζ) patahakan me owyownner: K oxvi? patasxan, ee ξ, η, ζ linen zowyg a

zowyg ankax: 383. Dicowq ξ1 ξ2 - mi nowyn erkraa akan baxowm owneco ankax patahakan me owyownner en: Apacowcel, or P (ξ1 = k/ξ1 + ξ2 = n) =

, n+1

k = 0, 1, 2, ..., n :

384. Dicowq ξ1 ξ2 - hamapatasxanabar λ1 λ2 paramet{ rerov Powasoni baxowm owneco ankax patahakan me owyown{ ner en: Cowyc tal, or P (ξ1 = k/ξ1 + ξ2 = n) = Cnk pk q n−k ,

k = 0, 1, ..., n,

orte p = λ1 /(λ1 + λ2 ), q = 1 − p: 385. Dicowq ξ η-n n p parametrerov ankax binomakan patahakan me owyownner en: Gtnel ξ + η = m paymani depqowm ξ patahakan me owyan baxowm: 386. Netowm en erkow za : Nkaragrel tarrakan patahowy{ neri tara owyown: Dicowq ξ -n a ajin za i vra <6> bacvelow ivn , η-n` <6> bacvelow ivn erkrord za i vra: Gtnel ξ η-i hamate baxowm: Apacowcel ξ η patahakan me owyownneri ankaxowyown: 387. Netowm en erkow za : Nkaragrel tarrakan patahowy{ neri tara owyown: Dicowq ξ -n a ajin za i vra bacva mia{ vorneri ivn , η-n` erkrord za i vra bacva miavorneri iv: Gtnel (ξ; η) vektori baxman renq: Apacowcel ξ η pataha{ kan me owyownneri ankaxowyown: 388. (ξ, η) patahakan vektor havasaraa baxva

A(−1, 0), B(1, 0), C(0, 1), D(0, −1) gaganerov qa akowsow nersowm: Gtnel fξ,η (x, y), fξ (x) fη (x) xtowyownner: Ankax e?n ardyoq ξ η patahakan me owyownner: 389. (ξ, η, ζ) patahakan vektor havasaraa baxva

glanowm, ori barrowyown havasar 2H , himqi a avi` R, kentron gtnvowm koordinatneri skzbnaketowm, isk nord zow{ gahe OZ a ancqin: Gtnel ayd vektori yowraqanyowr proyek{ ciayi baxman xtowyown: Ankax e?n ardyoq ayd proyekcianer:

390. (ξ, η, ζ) patahakan vektor havasaraa baxva

r a avov S gndi mej: a) Gtnel ayd vektori yowraqanyowr kom{ ponenti baxman xtowyown, b) i?n havanakanowyamb (ξ, η, ζ) patahakan ket kgtnvi S -i het hamakentron r/2 a avov gndi nersowm: 391. Apacowcel, or ee Fξ (x) Fη (y) baxman fownkcianer anndhat en hamapatasxanabar x0 y0 keterowm, apa erk{ a Fξ,η (x, y) baxman fownkcian anndhat (x0 , y0 ) ketowm: 392. Ka owcel aynpisi erka baxman fownkciayi rinak, or lini anndhat (x0 , y0 ) ketowm, sakayn mek a ani Fξ (x) Fη (y) baxman fownkcianer hamapatasxanabar x0 y0 ke{ terowm linen xzvo: 393. Dicowq F (x)- ξ patahakan me owyan baxman fownk{ cian : Gtnel a) (ξ, ξ), b) (ξ, |ξ|) patahakan vektori F (x, y) baxman fownkcian: 394. Dicowq ξ yan fownkcian `

η patahakan me owyownneri hamate xtow{

f (x, y) =

6 ( 2 xy ) x + ,

0 < x < 1, 0 < y < 2 :

a) Hamozvel, or ayn iskapes xtowyan fownkcia : b) Havel ξ patahakan me owyan xtowyan fownkcian: g) Gtnel P (ξ > η): d) Gtnel P (η > 12 /ξ < 12 ): 395. Dicowq ξ1 - ξ2 - Koii standart baxowm owneco an{ ξ1 +ξ2 kax patahakan me owyownner en: Apacowcel, or η = 1−ξ pa{ 1 ξ2 tahakan me owyown nowynpes owni Koii baxowm: 396. Dicowq ξ1 , ξ2 , . . . , ξn , . . . ankax miatesak baxva pa{ tahakan me owyownner en, oronq 1/2 havanakanowyownnerov n{ ∞ ∑ downowm en 0 kam 1 areqner: Gtnel η = 2ξkk patahakan me ow{ k=1 yan baxowm:

397. Dicowq ξ1 , ξ2 , . . . , ξn ankax, miatesak baxva pataha{ kan me owyownner en, nd orowm Gtnel

P (ξi = 1) = P (ξi = −1) = 21 , i = 1, 2, . . . , n : n ∏ ξi patahakan me owyan baxowm: ηn = i=1

398. Dicowq trva (ξ, η) erka patahakan me owyan ha{ mate baxowm` p(1, 1) = 0, 5, p(2, 1) = 0, 1, p(1, 2) = 0, 1, p(2, 2) = 0, 3 : Gtnel ξ patahakan me owyan baxowm η = 1 paymani depqowm:

399. Dicowq trva (ξ, η) erka patahakan me owyan ha{ mate xtowyan fownkcian {

f (x, y) =

c(y 2 − x2 )e−y , 0,

−y ≤ x ≤ y,

0<y<∞

mnaca depqerowm:

a) Gtnel c gor akic: b) Gtnel ξ η patahakan me owyownneri xtowyan fownk{ cianer: 400. Dicowq trva (ξ, η) erka patahakan me owyan ha{ mate xtowyan fownkcian {

f (x, y) =

Gtnel P (ξ < η)

e−(x+y) , 0,

P (ξ < a),

0 ≤ x < ∞,

0≤y<∞

mnaca depqerowm:

orte a > 0 hastatown iv :

401. Dicowq trva (ξ, η) erka patahakan me owyan ha{ mate xtowyan fownkcian { xe−x(1+y) , f (x, y) = 0,

η -i

x > 0,

y>0

mnaca depqerowm:

a) Gtnel ξ -i xtowyan fownkcian η = y paymani depqowm xtowyan fownkcian ξ = x paymani depqowm:

b) Gtnel ζ = ξη-i xtowyan fownkcian: 402. Dicowq trva (ξ, η) erka patahakan me owyan ha{ mate baxowm` p(1, 1) = 1/8,

p(2, 1) = 1/4,

p(1, 2) = 1/8,

p(2, 2) = 1/2 :

a) Gtnel η = i, i = 1, 2 paymani depqowm me owyan baxowm: b) Ankax e?n ardyoq ξ -n η-n: g) Havel P (ξη ≤ 3), P (ξ + η > 2), P (ξ/η > 1):

patahakan

403. Dicowq trva (ξ, η) erka patahakan me owyan ha{ mate xtowyan fownkcian {

f (x, y) =

c(x2 − y 2 )e−x , 0,

0 ≤ x < ∞,

−x ≤ y ≤ x

mnaca depqerowm: Gtnel ξ = x paymani depqowm η patahakan me owyan baxowm:

Patahakan me owyownneric fownkciayi baxowm Dicowq ξ = (ξ1 , ξ2 , . . . , ξn )-n n-a ani patahakan me ow{ yown , Fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn )- nra baxman fownkcian , isk η = (η1 , η2 , . . . , ηm )-n m -a ani patahakan me owyown , orte ηk = fk (ξ1 , ξ2 , . . . , ξn ), k = 1, 2, . . . , m Φη1 ,...,ηm (y1 , y2 , . . . , ym )-n nra baxman fownkcian : Ayd depqowm tei owni nerkayacowm` ∫

Φη1 ,η2 ,...,ηm (y1 , y2 , . . . , ym ) =

∫

dFξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ),

orte D = {(x1 , x2 , . . . , xn ) : fj (x1 , x2 , . . . , xn ) < yj , j = 1, 2, . . . , m} :

Ee n-a ani ξ = (ξ1 , ξ2 , . . . , ξn ) patahakan me owyownn owni fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn ) baxman xtowyown, apa Φη1 ,η2 ,...,ηm (y1 , y2 , . . . , ym ) =

∫ =

∫

fξ1 ,ξ2 ,...,ξn (x1 , x2 , . . . , xn )dx1 dx2 . . . dxn :

Ee ξ = (ξ1 , ξ2 , . . . , ξn )- n-a ani diskret patahakan me{

owyown , apa ∑

Φη1 ,η2 ,...,ηm (y1 , y2 , . . . , ym ) =

pi1 ,i2 ,...,in ,

i1 ,i2 ,...,in , fj (ki1 ,ki2 ,...,kin )<yj , j=1,2,...,m

orte pi1 ,i2 ,...,in = P (ξ1 = ki1 , ξ2 = ki2 , . . . , ξn = kin ) :

404. Dicowq ξ1 , ξ2 , ..., ξn ankax patahakan me owyownnern ownen mi nowyn F (x) baxman fownkcia: Gtnel ξ = min(ξ1 , ξ2 , ..., ξn )

η = max(ξ1 , ξ2 , ..., ξn )

patahakan me owyownneri baxman fownkcianer: 405. Dicowq ξ1 , ξ2 , ..., ξn patahakan me owyownnern ankax en ownen hamapatasxanabar λ1 , λ2 , ..., λn parametrerov cowc{ ayin baxowm: Gtnel η = min(ξ1 , ξ2 , ..., ξn ) patahakan me owyan baxman fownkcian: 406. Dicowq haytni en ξ ri baxman renqner` ξ P

Gtnel ξ + η, renqner:

−1 0, 2

0, 3

0, 5

ξ − η, ξ · η

ankax patahakan me owyownne{ η P

0, 4

0, 6

patahakan me owyownneri baxman

407. Dicowq ξ η ankax patahakan me owyownnern ownen het yal baxman renqner`

ξ P

Gtnel ξ + η, renqner:

0, 2

0, 4

ξ − η, ξ · η

η P

0, 3

0, 4

patahakan me owyownneri baxman

408. ξ1 ξ2 ankax patahakan me owyownner baxva en mi nowyn erkraa akan renqov (P (ξ = k) = qk p, k = 0, 1, 2, ...): Dicowq η = max(ξ1 , ξ2 ): Gtnel η-i baxowm (η, ξ1 )-i baxowm: 409. Dicowq ξ1 , ξ2 , ..., ξn - mi nowyn F (x) baxman fownkciayov ankax patahakan me owyownner en: Nanakenq ξ = min(ξ1 , ξ2 , ..., ξn )

η = max(ξ1 , ξ2 , ..., ξn ) :

Gtnel (ξ, η) patahakan vektori baxman fownkcian gtagor{

elov hamate baxman fownkcian, gtnel ξ η-i miaa bax{ man fownkcianer: 410. Dicowq ξ1 ξ2 ankax patahakan me owyownnern ownen hamapatasxanabar λ1 λ2 parametrerov cowcayin baxowm: Apacowcel η1 = ξ1 −ξ2 η2 = min(ξ1 , ξ2 ) patahakan me owyownne{ ri ankaxowyown: 411. Dicowq (ξ1 , ξ2 ) patahakan ketn owni havanakanowyown{ neri havasaraa baxowm {(x, y); 0 ≤ x ≤ a, 0 ≤ y ≤ a} qa akowsow nersowm: Cowyc tal, or ξ1 −ξ2 min(ξ1 , ξ2 ) patahakan me owyownner ownen mi nowyn baxowm, aysinqn, cankaca t-i ha{ mar P (ξ1 − ξ2 < t) = P (min(ξ1 , ξ2 ) < t): 412. Dicowq ξ1 ξ2 - anndhat baxman fownkcianerov ankax patahakan me owyownner en η = ξ1 + ξ2 : Apacowcel, or a)

∫∞

∫∞ Fξ2 (x − u)dFξ1 (u) =

Fη (x) = −∞

Fξ1 (x − v)dFξ2 (v); −∞

b) ee ξ1 ξ2 patahakan me owyownneric mekn owni havanaka{ nowyownneri baxman xtowyown, apa ∫∞ fξ1 (x − v)dFξ2 (v)

fη (x) =

kam fη (x) =

∫∞ fξ2 (x − u)dFξ1 (u); −∞

−∞

g) ee ξ1 -, yownner, apa ′

′

ξ2 - ownen havanakanowyownneri baxman xtow{

∫∞

∫∞ fξ1 (x − v)fξ2 (v) dv =

fη (x) = −∞

fξ2 (x − u)fξ1 (u) du : −∞

413. Dicowq ξ1 ξ2 - anndhat baxman fownkcianerov ankax patahakan me owyownner en η = ξ1 − ξ2 : Apacowcel, or a)

∫∞ Fη (x) =

∫∞ [1 − Fξ2 (u − x)]dFξ1 (u);

Fξ1 (x + v)dFξ2 (v) = −∞

−∞

b) ee ξ1 ξ2 patahakan me owyownneric mekn owni baxman xtowyown, apa η-n s owni xtowyown ∫∞ fη (x) =

fξ1 (x + v)dFξ2 (v)

kam fη (x) =

−∞

g) ee

′

∫∞ fξ2 (u − x)dFξ1 (u); −∞

ξ1 -,

′

ξ2 -

ownen baxman xtowyownner, apa

∫∞ fη (x) =

∫∞ fξ2 (u − x)fξ1 (u) du :

fξ1 (x + v)fξ2 (v) dv = −∞

−∞

414. Dicowq ξ1 ξ2 ankax patahakan me owyownnern ownen anndhat baxman fownkcianer η = ξ1 · ξ2 : Apacowcel, or a)

∫0 [ ∫∞ ( x )] (x) Fη (x) = 1 − Fξ2 dFξ1 (u) + Fξ2 dFξ1 (u), u u −∞

b) ee ξ1 ξ2 patahakan me owyownneric mekn owni havanaka{ nowyownneri baxman xtowyown, apa η-n s kownena havana{ kanowyownneri baxman xtowyown ∫∞ fη (x) = −∞

g) ee apa

ξ1

(x) fξ1 dFξ2 (v) |v| v ξ2

kam fη (x) =

∫∞ −∞

(x) fξ2 dFξ1 (u) : |u| v

patahakan me owyownnern ownen xtowyownner, ∫∞ fη (x) = −∞

(x) fξ2 (u) du : fξ1 |u| u

415. Dicowq ξ1 ξ2 ankax patahakan me owyownnern ownen anndhat baxman fownkcianer η = ξξ12 : Apacowcel, or a) Fη (x) =

∫0

∫∞ [1 − Fξ1 (vx)] dFξ2 (v) +

−∞

Fξ1 (vx) dFξ2 (v),

b) ee ξ1 -n owni xtowyown, apa η-n s kownena xtowyown ∫∞ |v| · fξ1 (vx)dFξ2 (v),

fη (x) = −∞

g) ee ξ1 -

ξ2 -

ownen xtowyownner, apa ∫∞ |v|fξ1 (vx)fξ2 (v) dv :

fη (x) = −∞

416. Dicowq fξ1 ,ξ2 (x, y)- (ξ1 , ξ2 ) patahakan vektori hava{ nakanowyownneri baxman xtowyownn : Apacowcel, or ξ1 + ξ2 ξ1 −ξ2 patahakan me owyownnern ownen het yal havanakanow{ yownneri baxman xtowyownner` a) fξ1 +ξ2 (z) =

∫∞

∫∞ fξ1 ,ξ2 (x, z − x) dx =

−∞

fξ1 ,ξ2 (z − y, y) dy, −∞

∫∞

b) fξ1 −ξ2 (z) =

∫∞ fξ1 ,ξ2 (x, x − z) dx :

fξ1 ,ξ2 (z + y, y) dy = −∞

−∞

417. Dicowq fξ1 ,ξ2 (x, y)- (ξ1 , ξ2 ) patahakan vektori hava{ nakanowyownneri baxman xtowyownn : Apacowcel, or ξ1 · ξ2 ξ1 ξ2 patahakan me owyownnern ownen het yal havanakanowyown{ neri baxman xtowyownner` (

∫∞

a) fξ1 ·ξ2 (z) =

fξ1 ,ξ2 −∞

b) fξ1 /ξ2 (z) =

z ,y y

)

dy = |y|

∫∞ −∞

( z ) dx fξ1 ,ξ2 x, , x |x|

∫∞ fξ1 ,ξ2 (zy, y) · |y| dy : −∞

418. Dicowq ξ η ankax patahakan me owyownnern ndownowm en 0, 1, ..., n areqner, nd orowm P (ξ = i) = P (η = i) = n+1 , i = 0, ..., n: Gtnel ξ + η patahakan me owyan baxman renq: 419. ξ1 ξ2 ankax patahakan me owyownnern ownen Powasoni baxowm hamapatasxanabar λ1 λ2 parametrerov: Apacow{ cel, or η = ξ1 + ξ2 patahakan me owyown owni Powasoni baxowm λ1 + λ2 parametrov: 420. Dicowq ξ η-n ankax, miatesak baxva amboj areq{ ner ndowno patahakan me owyownner en, pi = P (ξ = i), i = 0, ±1, ±2, ...: Apacowcel, or P (ξ − η = 0) =

∞ ∑

p2i :

i=−∞

421. Dicowq ξ η-n ankax patahakan me owyownner en mi nowyn xtowyan fownkciayov: Apacowcel, or

f (x)

∫∞ f 2 (x) dx :

fξ−η (0) = −∞

422. Dicowq ξ1 ξ2 - [a, b] mijakayqowm havasaraa baxva

patahakan me owyownner en: Gtnel η = ξ1 + ξ2 patahakan me owyan havanakanowyownneri baxman xtowyown: 423. Dicowq ξ -n η-n [−a/2, a/2] mijakayqowm havasaraa

baxva ankax patahakan me owyownner en: Gtnel a) η = ξ1 + ξ2 , b) η = ξ1 − ξ2 patahakan me owyan havanakanowyown{ neri baxman xtowyown: 424. Dicowq ξ1 ξ2 - [0, 1] mijakayqowm havasaraa baxva

patahakan me owyownner en: Gtnel a) η = ξ1 + ξ2 , b) η = ξ1 − ξ2 , g) η = |ξ1 − ξ2 |, d) η = ξ1 · ξ2 , e) η = ξ1 /ξ2 patahakan me owyown{ neri havanakanowyownneri baxman xtowyown: 425. Gtnel ξ η patahakan me owyownneri (ξ + η) gowmari havanakanowyownneri baxman xtowyown, ee ξ -n [0, 2] hat{ va owm havasaraa baxva patahakan me owyown , isk η-n havasaraa baxva [−1, 1] hatva owm: 426. Dicowq ξ1 - ξ2 - ankax, miatesak baxva pata{ hakan me owyownner en, oronc havanakanowyownneri baxman xtowyown havasar f (x) = 12 e−|x| : Gtnel η = ξ1 + ξ2 pataha{ kan me owyan havanakanowyownneri baxman xtowyown: 427. Dicowq ξ η patahakan me owyownnern ankax en, nd orowm ξ -n owni cowcayin baxowm λ parametrov, isk η-n havasa{ raa baxva [0, a] mijakayqowm: Gtnel ξ + η patahakan me owyan havanakanowyownneri baxan xtowyown: 428. Dicowq ξ1 ∼ N (a1 , σ12 ) ξ2 ∼ N (a2 , σ22 ) ankax patahakan me owyownnern ownen normal baxowmner: Apacowcel, or η = ξ1 + ξ2 patahakan me owyown s owni N (a1 + a2 , σ12 + σ22 ) normal ba{ xowm: 429. Dicowq ξ1 ξ2 ankax patahakan me owyownnern ownen hamapatasxanabar (α, β1 ) (α, β2 ) parametre{

Γ-baxowm

rov: Apacowcel, or η = ξ1 + ξ2 patahakan me owyown s owni Γ-baxowm (α, β1 + β2 ) parametrerov: 430. Dicowq ξ1 ξ2 ankax patahakan me owyownnern ownen cowcayin baxowm λ parametrov: Oroel a) η = ξ1 − ξ2 , b) η = ξ1 /ξ2 patahakan me owyownneri baxman xtowyown: 431. ξ η patahakan me owyownnern ankax en, nd orowm fξ (x) = 12x2 (1 − x), x ∈ (0, 1), isk fη (y) = 2y , y ∈ (0, 1): Gtnel ξ · η patahakan me owyan xtowyan fownkcian:

432. Dicowq ξ -n η-n ankax, N (0, σ) normal baxowm owneco patahakan me owyownner en: Apacowcel, or ζ = ξ/η patahakan me owyown baxva Koii renqov: 433. (ξ1 , ξ2 ) patahakan ket havasaraa baxva r = 1 a avov rjani mej: Gtnel η = ξ2 /ξ1 patahakan me owyan havanakanowyownneri baxman xtowyown: 434. ξ patahakan me owyownn owni Koii baxowm: Apacowcel, 3ξ−ξ 3 2ξ , g) patahakan me owyown s owni Koii or a) 1/ξ , b) 1−ξ 1−3ξ 2 baxowm: 435. Gtnel η = ξ1ξ+ξ patahakan me owyan baxman fownk{ cian, ee ξ1 ξ2 patahakan me owyownnern ankax en ownen havanakanowyownneri cowcayin baxowm` f (x) = e−x , x ≥ 0: 436. Gtnel η = ξ1ξ+ξ patahakan me owyan havanakanow{ yownneri baxman xtowyown, ee ξ1 ξ2 patahakan me owyown{ nern ankax en ownen havanakanowyownneri cowcayin baxowm` f (x) = e−x , x ≥ 0: 437. Gtnel η = ξ1ξ+ξ patahakan me owyan havanakanow{ yownneri baxman xtowyown, ee ξ1 ξ2 patahakan me owyown{ nern ankax en havasaraa baxva [0, 1] mijakayqowm:

438. (ξ1 , ξ2 ) patahakan vektorn owni {

fξ1 ,ξ2 (x, y) =

x + y, 0,

0 ≤ x ≤ 1, 0 ≤ y ≤ 1

mnaca depqerowm

havanakanowyownneri baxman xtowyown: Gtnel η = ξ1 + ξ2 patahakan me owyan havanakanowyownneri baxman xtow{ yown: 439. (ξ, η) patahakan vektorn owni {

f (x, y) =

24xy(1 − x)2 , 0,

0 < x < 1, 0 < y < 1

mnaca depqerowm

havanakanowyownneri baxman xtowyown: Gtnel (ξ · η) pata{ hakan me owyan havanakanowyownneri baxman xtowyown: 440. (ξ, η) patahakan vektori havanakanowyownneri bax{ man xtowyown havasar {

f (x, y) =

xe−x(1+y) , 0,

x > 0, y > 0

mnaca depqerowm :

Oroel (ξ·η) patahakan me owyan havanakanowyownneri bax{ man xtowyown: 441. Dicowq ξ1 ξ2 - standart normal` N (0, 1) baxowm owneco, ankax patahakan me owyownner en: Apacowcel, or ξ1 − ξ2 ξ1 + ξ2 patahakan me owyownnern ankax en: 442. ξ η patahakan me owyownnern ankax en ownen cowc{ ayin baxowm λ = 1 parametrov: Apacowcel (ξ +η) ξ/η pata{ hakan me owyownneri ankaxowyown: 443. Dicowq ξ1 - ξ2 - ankax standart normal` N (0, 1) ba{ xowm owneco patahakan me owyownner en: Havel P (ξ12 + ξ22 < R 2 ):

444. Dicowq ξ1 ξ2 - standart normal` N (0, 1) baxva , ankax patahakan me owyownner en, isk θ-n havasaraa baxva

[0, 2π] mijakayqowm: Gtnel ξ1 cos θ + ξ2 sin θ patahakan me owyan baxowm: 445. Dicowq ξ1 , ξ2 , ..., ξn ankax, miatesak baxva pataha{ kan me owyownnern ownen cowcayin baxowm λ > 0 parametrov: n ∑ ξk patahakan me owyan havanaka{ Apacowcel, or Sn = k=1 nowyownneri baxman xtowyownn owni het yal tesq` { fSn (x) =

(λx)n−1 −λx , (n−1)! λe

x≥0

x<0:

446. Dicowq ξ1 , ξ2 , ..., ξn - standart normal` N (0, 1) baxva , n ∑ ankax patahakan me owyownnern en: Apacowcel, or χ2 = ξk2 k=1 patahakan me owyown owni  

fχ2 (x) =

0,

x 2 −1 e− 2 , n

2n/2 Γ n

( )

x ≥ 0, x<0

havanakanowyownneri baxman xtowyown: 447. Dicowq ξ, ξ1 , ξ2 , ..., ξn patahakan me owyownnern ankax en ownen standart normal` N (0, 1) baxowm: Apacowcel, or ξ

η=√ n

n ∑ i=1

ξi2

patahakan me owyan havanakanowyownneri baxman xtow{ yown havasar ) ( Γ n+1 2 − n+1 2( ) fη (x) = √ : n (1 + x ) πΓ 2

Patahakan me owyan vayin bnowagriner patahakan me owyan maematikakan spasowm kam mijin areq anvanowm en het yal iv` ξ = ξ(ω)

∫

Eξ =

ξ(ω)P (dω), Ω

ee aj masowm grva Lebegi integral goyowyown owni: Ayn hamnk{ nowm x-i Stiltyesi integrali het ∫∞ Eξ =

xdFξ (x), −∞

ee integral bacarak zowgamet : Diskret ξ patahakan me owyan maematikakan spasowm havasar ` Eξ =

∑

x k pk ,

∑

|xk |pk < ∞ :

Bacarak anndhat ξ patahakan me owyan maematikakan spasowm havasar ` ∫∞ xfξ (x)dx,

Eξ =

−∞

g(x)

∫∞ |x|fξ (x)dx < ∞ : −∞

anndhat fownkciayi hamar ∫ Eg(ξ) =

∫∞ g(ξ(ω))P (dω) =

g(x)dFξ (x) : −∞

Ω

Diskret ξ patahakan me owyan depqowm Eg(ξ) =

∑ k

g(xk )pk ,

∑ k

|g(xk )|pk < ∞ :

Bacarak anndhat ξ patahakan me owyan depqowm ∫∞ Eg(ξ) =

g(x)fξ (x)dx, −∞

∫∞

|g(x)|fξ (x)dx < ∞ : −∞

iv anvanowm en ξ patahakan me owyan k-rd kargi skzbnakan moment (k-n irakan iv ), µk = E(ξ − Eξ)k iv anvanowm en ξ -i k-rd kargi kentronakan moment (k-n irakan iv ): Erkrord kargi kentronakan moment √ anvanowm en dispersia nanakowm en Dξ = E(ξ − Eξ) : σ = Dξ me owyown anvanowm en mijin qa akowsayin eowm: ξ η patahakan me owyownneri korelyaciayi gor akic an{ vanowm en het yal me owyown` νk = Eξ k

ρ(ξ, η) =

E(ξ − Eξ)(η − Eη) √ , DξDη

Dξ ̸= 0, Dη ̸= 0 :

E(ξ−Eξ)(η−Eη) me owyown anvanowm en kovariacia

nanakowm`

cov(ξ, η) = E(ξ − Eξ)(η − Eη) :

Nenq patahakan me owyownneri vayin bnowagrineri hat{ kowyownner: 1. EC = C , C -n hastatown : 2. ECξ = CEξ , C -n hastatown : 3. Eξ ≥ 0, ee P (ξ ≥ 0) = 1: 4. Ee ξ ≥ 0 Eξ = 0, apa P (ξ = 0) = 1: 5. Eξ ≥ Eη, ee P (ξ ≥ η) = 1: 6. EIA (ω){ = P (A), orte IA (ω)-n A bazmowyan indikatorn ` 1, ω ∈ A 0, ω ∈ /A E(ξ ± η) = Eξ ± Eη ,

IA (ω) =

7. ee goyowyown ownen nva ereq ma. spasowmneric gone erkows: 8. Eξη = EξEη, ee ξ η-n ankax patahakan me owyownner en:

9. DC = 0, orte C -n hastatown iv : 10. Ee Dξ = 0, apa P (ξ = C) = 1, orte C = Eξ : 11. D(aξ + b) = a2 Dξ , a-n b-n hastatownner en: 12. D(ξ ± η) = Dξ + Dη, ee ξ -n η-n ankax patahakan me owyownner en: n n ∑ ∑ ∑ 13. D( ξk ) = Dξk + 2 cov(ξk , ξj ): k=1

k=1

k<j

14. |ρ(ξ, η)| ≤ 1 : 15. |ρ(ξ, η)| = 1 ayn miayn ayn depqowm, erb P (aξ + bη = c) = 1, orte a-n, b-n, c-n hastatownner en: 16. Ee ξ -n η-n ankax en, apa ρ(ξ, η) = 0: Haka ak it : Ee ρ(ξ, η) = 0, apa ξ -n η-n kovowm en korelacva :

Patahakan vektori vayin bnowagriner ξ = (ξ1 , ξ2 , . . . , ξn ) pahatakan vektori maematikakan spa{ sowm anvanowm en Eξ = (a1 , a2 , . . . , an ) vektor, orte ak = Eξk , k = 1, 2, . . . , n : ξ = (ξ1 , ξ2 , . . . , ξn ) pahatakan vektori dispersion (kova{

riacion) matric kam dispersia anvanowm en het yal matric` 

b11 b12  b21 b22 B=  ... ... bn1 bn2

 . . . b1n . . . b2n  , ... ...  . . . bnn

orte bij = cov(ξi , ξj ) = E(ξi − Eξi )(ξj − Eξj ); i, j = 1, 2, . . . , n : Nenq B matrici hatkowyownner` 1. bij = bji ; i, j = 1, 2, . . . , n` matric hamaa : 2. bii = Dξi ; i = 1, 2, . . . , n : 3. Cankaca α1 , α2 , . . . , αn irakan veri hamar tei owni n n ∑ ∑

bij αi αj ≥ 0,

i=1 j=1

aysinqn B matric o bacasakan orova :

Nenq oro kar or anhavasarowyownner, orte masnakcowm en patahakan me owyownneri momentner: Koow-Bownyakovskow anhavasarowyown` ee ξ η pata{ hakan me owyownnern aynpisin en, or Eξ 2 < ∞, Eη2 < ∞, apa E|ξη| ≤

√

Eξ 2

√

Eη 2 :

Lyapownovi anhavasarowyown` 0 < s < r depqowm (E|ξ|s )1/s ≤ (E|ξ|r )1/r :

Gyolderi anhavasarowyown` ee E|η|q < ∞,

E|ξ|p < ∞,

apa

p > 1, q > 1,

= 1,

E|ξη| ≤ (E|ξ|p ) p (E|η|q ) q :

Minkovskow anhavasarowyown` ee E|ξ|r < ∞, E|η|r < ∞, r ≥ 1,

apa

(E|ξ + η|r ) r ≤ (E|ξ|r ) r + (E|η|r ) r :

Paymanakan maematikakan spasowm Dicowq ξ η-n diskret patahakan me owyownner en: ξ pa{ tahakan me owyan paymanakan maematikakan spasowm η = y paymani depqowm kovowm E(ξ/η = y) =

∑

xP (ξ = x/η = y) =

∑

xpξ/η (x/y)

me owyown, orte pξ/η (x/y) =

Ee ξ apa

η -n

P (ξ = x, η = y) , Pη (y)

Pη (y) = P (η = y) > 0 :

bacarak anndhat patahakan me owyownner en, ∫

∞

E(ξ/η = y) = −∞

xfξ/η (x/y) dx,

orte fξ/η (x/y) =

fξ,η (x, y) , fη (y)

fη (y) > 0

paymanakan havanakanayin baxman xtowyownn :

448. ξ patahakan me owyown ndownowm 0, ±1, ±2, ..., ±n ar{ eqner P (ξ = i) = 2n+1 havanakanowyownnerov: Gtnel Eξ -n Dξ -n: 449. Nanaketin krakowm en 20 angam: Mek krakocov dipelow havanakanowyown havasar 0,7-i: Gtnel dipowmneri vi ma. spasowm dispersian: 450. ξ patahakan me owyown owni binomakan baxowm n, p parametrerov: Haytni, or Eξ = 12, Dξ = 4: Gtnel n- p-n: 451. Xaaco ahowm 30$, ee netva ereq za eric yowra{ qanyowri vra bacvowm <6>-, ahowm 20$, ee drancic erkowsi vra bacvowm <6>-, 10$, ee miayn meki vra bacvowm <6>-: Gtnel xaacoi aha gowmari ma. spasowm: 452. 10 artadranqneric 3- xotan en: Patahakanoren verc{ nowm en erkow artadranq: Gtnel dranc mej gtnvo xotan ar{ tadranqneri vi mijin dispersian: 453. 2 spitak 3 s gndik parownako sa oric pataha{ kanoren hanowm en 2 gndik: Gtnel dranc mej spitak gndikneri vi ma. spasowm dispersian: 454. A patahowyi i hayt galow havanakanowyown n ankax

oreric yowraqanyowrowm havasar p-i: Gtnel A patahowyi i hayt galow i hayt galow veri tarberowyan ma. spasowm:

455. ξ patahakan me owyownn ndownowm o bacasakan am{ boj areqner, nd orowm Eξ < +∞: Apacowcel, or Eξ =

n ∑

P (ξ ≥ i) :

i=1

456. ξ patahakan me owyown ndownowm o bacasakan am{ boj n ≥ 0 areqner n k n! A k -n,

pn = A

havanakanowyownnerov: Gtnel

ee haytni , or Eξ = a:

457. ξ patahakan me owyownn ndownowm o bacasakan am{ boj areqner P (ξ = n) =

an , (1 + a)n+1

a>0

havanakanowyownnerov (Paskali baxowm): Havel Eξ -n

Dξ -n:

458. ξ patahakan me owyownn ndownowm drakan amboj ar{ eqner nvazo erkraa akan progresia kazmo havanaka{ nowyownnerov: Gtnel ayd progresiayi a ajin a andam q hay{ tararn aynpes, or Eξ = 10: 459. ξ patahakan me owyownn ndownowm 0, 1, ... areqner nvazo erkraa akan progresia kazmo havanakanowyownne{ rov: a) Gtnel Eξ -i Dξ -i mij ea kaxva owyown: b) Gtnel P (ξ = n), n = 0, 1, ..., ee haytni , or Eξ = a: 460. Metaadram netowm en min <gerbi> a ajin angam i hayt gal: Gtnel netowmneri vi ma. spasowm dispersian: 461. Ankax oreric yowraqanyowrowm A patahowyi i hayt galow havanakanowyown havasar p-i (0 < p < 1): orer katarowm en min A patahowyi a ajin angam i hayt gal: Gtnel katarvo oreri ξ vi ma. spasowm dispersian:

462. [a, b] mijakayqowm havasaraa baxva ξ patahakan me owyan ma. spasowm havasar Eξ = 4, isk dispersian` Dξ = 3: Gtnel ξ patahakan me owyan baxman xtowyown: 463. ξ patahakan me owyown baxva

f (x) = ae−λ|x| , λ > 0: Gtnel a, Eξ , Dξ -n:

Laplasi renqov`

464. ξ patahakan me owyown baxva elei renqov` {

f (x) =

0, 2 2 Axe−λ x ,

x≤0 x>0:

Gtnel A, Eξ , Dξ -n: 465. ξ patahakan me owyownn owni {

f (x) =

xn −x n! e ,

x≥0 x<0

havanakanowyownneri baxman xtowyown: Gtnel Eξ

Dξ -n:

466. ξ patahakan me owyownn owni het yal baxman renq` ξ -1 0 1 P 1/3 1/3 1/3 Gtnel a) η = |ξ| patahakan me owyan baxowm, b) Eη Dη-n: 467. ξ patahakan me owyownn owni het yal baxman renq` ξ -1 0 1 2 P 0,2 0,1 0,3 0,4 Gtnel η = 2ξ patahakan me owyan ma. spasowm disper{ sian: 468. Dicowq ξ patahakan λ parametrov: Gtnel η = e−ξ sowm dispersian:

me owyownn owni cowcayin baxowm patahakan me owyan ma. spa{

469. ξ patahakan me owyown havasaraa baxva

[0, 1] mijakayqowm: Gtnel a) E sin2 πξ , b) Eeξ : 470. ξ patahakan me owyownn owni het yal baxman xtow{ yown` { [ ] f (x) =

cos x, 0,

Gtnel a) η = sin ξ , b) η spasowm dispersian: 471. Dicowq Eξ = 0 b) E min(0, ξ):

x ∈ − π2 , π2 [ ]: x∈ / − π2 , π2

= | sin ξ|

E|ξ| = 1:

patahakan me owyan ma.

Gtnel a) E max(0, ξ),

472. Dicowq ξ patahakan me owyan baxman xtowyown ha{ vasar f (x) =

: π(1 + x2 )

Havel E min(|ξ|, 1): 473. l erkarowyown owneco o patahakanoren kotrel en 2 masi: Gtnel oqr masi η erkarowyan baxman fownkcian, Eη-n Dη -n: 474. Erka (ξ, η) patahakan me owyan havanakanowyown{ neri baxman renqn ` η\ξ 0 -1 0,1 0,2 0 0,2 0,3 0 0,2 Gtnel η = 2ξ + η2 patahakan me owyan ma. spasowm dispersian: 475. Trva erka patahakan me owyan havanakanow{ yownneri baxman ayowsak`

η\ξ

0,01 0,05 0,12 0,02 0 0,01 0,02 0 0,01 0,05 0,02 0,02 0 0,05 0,1 0 0,3 0,05 0,01 0 0,02 0,01 0,03 0,1

Gtnel` 1. P (ξ = 2/η = 3), 3. E(ξ + η), 5. P (ξ + η < 5/η ≤ 2), 2. E(ξ/η = 1), 4. E(ξ 2 /η ≤ 1), 6. E(ξη/η ≤ 1): 476. Netowm en erkow za : Dicowq ξ -n bacva miavorneri ivn a ajin za i vra, isk η-n bacva erkow miavorneric me agowynn : a) Gtnel ξ η patahakan me owyownneri hamate baxowm, b) havel Eξ , Dξ , Eη, Dη cov(ξ, η)-n: 477. Gtnel ζ = 2ξ − 3η patahakan me owyan ma. spasowm dispersian, ee Eξ = 0, Eη = 2, Dξ = 2, Dη = 1, ρ(ξ, η) = − √ :

478. Nanakenq erkow za eri vra bacva miavorneri gowmar tarberowyown hamapatasxanabar ξ -ov η-ov: Apacowcel, or ξ η patahakan me owyownner ankax en: 479. ξ η patahakan me owyownnern ankax en normal baxva mi nowyn (a, σ2 ) parametrerov: Gtnel v1 = αξ + βη v2 = αξ−βη patahakan me owyownneri korelyaciayi gor akic: 480. Dicowq ξ1 ξ2 - ankax mi nowyn verjavor dispersia owneco patahakan me owyownner en: Apacowcel, or η1 = ξ1 + ξ2 η1 = ξ1 − ξ2 patahakan me owyownner korelacva en: 481. ξ patahakan me owyown havasaraa baxva [0, 1] hatva owm: Gtnel het yal patahakan me owyownneri korelya{ ciayi gor akic` a) ξ ξ 2 ,

b) ξ

ξ3:

482. ξ patahakan me owyown havasaraa baxva [−1, 1] hatva owm: Gtnel het yal patahakan me owyownneri korelya{ ciayi gor akic` a) ξ sin πξ2 , b) sin πξ2 cos πξ2 : 483. Dicowq ξ η-n ankax patahakan me owyownner en, nd orowm Eξ = 1, Eη = 2, Dξ = 1, Dη = 4: Gtnel het yal patahakan me owyownneri ma. spasowmner` a) ξ 2 + 2η2 − ξη − 4ξ + η + 4, b) (ξ + η + 1)2 : 484. Dicowq ξ patahakan me owyownn owni het yal xtowyan fownkcian` { a) f (x) = b) f (x) =

{

0, ke−k(x−a) ,

x<a x ≥ a,

0, 1 − |x − 1|,

x∈ / [0, 2] x ∈ [0, 2] :

Havel ayd patahakan me owyownneri ma. spasowm disper{ sian: 485. Dicowq ξ η-n ndownowm en -1 kam 1 areqner het yal havanakanowyownnerov` p(i, j) = P (ξ = i, η = j),

i = −1, 1, j = −1, 1 :

Enadrenq

Eξ = Eη = 0: Apacowcel, or p(1, 1) = p(−1, −1) p(−1, 1) = p(1, −1) gtnel Dξ , Dη , cov(ξ, η):

486. Dicowq trva (ξ, η) erka patahakan me owyan ba{ xowm p(1, 1) = 1/9,

p(2, 1) = 1/3,

p(3, 1) = 1/9

p(1, 2) = 1/9, p(2, 2) = 0, p(3, 2) = 1/18 p(1, 3) = 0, p(2, 3) = 1/6, p(3, 3) = 1/9: a) Gtnel E(ξ/η = i) i = 1, 2, 3 : b) Ankax e?n ardyoq ξ -n η-n:

B 487. Sa or parownakowm N gndik, oroncic n- spitak : Hanel en m gndik (m ≤ min(n, N − n): Dicowq ξ -n hanva

gndikneri mej spitak gndikneri qanakn : Gtnel a) ξ pataha{ kan me owyan baxowm (ayn anvanowm en hipererkraa akan), b) gtnel Eξ -n Dξ -n: 488. 2 spitak 4 s gndik parownako sa oric hanowm en 3 gndik tea oxowm en erkrord sa or, orte kar 5 spitak gndik: Aynowhet erkrord sa oric tea oxowm en a ajin sa or 4 gndik: Oroel erkow sa ornerowm spitak gndikneri ξ1 ξ2 veri ma. spasowmner: 489. Dicowq n ankax orerowm, oroncic yowraqanyowrowm A patahowyi i hayt galow iv havasar µ-i, P (A) = p: ξ -n patahakan me owyown , orn ndownowm 0 kam 1 areqner` kaxva

µ-i zowyg kam kent linelowc: Gtnel Eξ -n: 490. m spitak n s gndik parownako sa oric hanowm en mekakan gndik, yowraqanyowr angam veradarnelov ayn sa or, min spitak gndiki a ajin angam i hayt gal: Gtnel hanva

s gndikneri vi ma. spasowm: 491. Dicowq ξ -n Be nowlii ankax oreri hajordakanowyan a ajin oric sksva <seriayi> (hajoowyownneri kam anha{ joowyownneri) erkarowyownn : Gtnel ξ patahakan me owyan baxowm, Eξ Dξ -n:

492. ξ patahakan me owyownn owni (α, β) parametrerov Γ-ba{ xowm, aysinqn nra baxman xtowyown havasar {

f (x) =

Gtnel Eξ

αβ β−1 −αx e , Γ(β) x

x ≥ 0,

x<0:

Dξ -n:

493. ξ patahakan me owyownn owni Powasoni baxowm λ pa{ rametrov: Havel E 1+ξ : 494. Dicowq F (x) baxman fownkcia owneco me owyownn owni ma. spasowm: Apacowcel, or ∫0 Eξ = −

patahakan

∫∞ F (x) dx + [1 − F (x)] dx :

−∞

495. Dicowq ξ patahakan me owyownn owni F (x) baxman fownk{ cia goyowyown owni nra ma. spasowm: Apacowcel, or tei ownen het yal a nowyownner` lim x(1 − F (x)) = 0,

x→+∞

496. yown`

lim x F (x) = 0 :

x→−∞

patahakan me owyownn owni het yal baxman xtow{ { f (x) =

Gtnel c hastatown sowm dispersian:

0, cx,

η = ξ2

x∈ / (0, 1) x ∈ (0, 1) :

patahakan me owyan ma. spa{

497. a) Gtnel E|ξ|-n, ee ξ patahakan me owyownn owni (0, σ2 ) parametrerov normal baxowm, b) Gtnel E|ξ−a|, ee ξ -n (a, σ2 ) parametrerov normal baxva

patahakan me owyown :

498. [0, a] hatva i vra patahakanoren nowm en erkow ket: Dicowq η-n nranc mij ea he avorowyownn : Oroel Eη-n Dη -n: 499. P ket havasaraa baxva (0, 0) kentronov R a avi owneco rjani mej: Dicowq P keti he avorowyown rja{ ni kentronic havasar η-i: Gtnel Eη-n Dη-n: 500. P ket havasaraa baxva (0, 0) kentronov R a avi owneco rjanag i vra: P ketic rjanin tarva oa{

o: Gtnel oa oi ayn hatva i ξ erkarowyan baxman fownk{ cian xtowyown, or miacnowm P ket 0X a ancqi nra hat{ man keti het: Goyowyown owni? ardyoq Eξ -n: 501. R a avi owneco rjanag i vra patahakanoren verc{ nowm en erkow ket: Gtnel dranc mij ea ξ he avorowyan bax{ man fownkcian havel Eξ -n: 502. Dicowq ξ η-n o bacasakan amboj areqner ndowno ankax patahakan me owyownner en, nd orowm Eξ < +∞: Apa{ cowcel, or E min(ξ, η) =

∞ ∑

P (ξ ≥ i) P (η ≥ i) :

i=1

503. Dicowq ξ η-n [0, 1] mijakayqowm havasaraa baxva

patahakan me owyownner en: Apacowcel, or ξ -i η-i cankaca

kaxva owyan depqowm E|ξ − η| ≤ 1/2: (Cowcowm` havel E|ξ − 1/2| E|η − 1/2| gtagor el |x − y| ≤ |x| + |y| anhavasarowyown:) 504. Dicowq ξ patahakan me owyown normal baxva (0, 1) parametrerov: Havel η = cos ξ patahakan me owyan ma. spasowm dispersian: 505. Dicowq ξ1 , ξ2 , ..., ξn drakan, miatesak baxva , ankax patahakan me owyownner en: Apacowcel, or ee k ≤ n, apa (

ξ1 + ξ2 + ... + ξk ξ1 + ξ2 + ... + ξn

)

k : n

506. 1, 2, ..., 29, 30 veric anveradar nmowahanman sxemayov vercnowm en tas iv: Gtnel nva veri gowmari maematikakan spasowm: (Cowcowm` dicowq ξk -n, k = 1, 2, ..., 10, k-rd nva ivn : Apacowcel, or ξk -er miatesak en baxva ): 507. Grva en n namakner, sakayn rarner hasceagrva

en patahakan kargov: Dicowq ξ -n ayn namakneri ivn , oronq hasel en irenc hasceatererin: Gtnel Eξ -n Dξ -n: 508. Apacowcel, or Eξη = EξEη havasarowyownic ndhanowr depqowm i bxowm ξ -i η-i ankaxowyown: 509. Apacowcel, or Eξη = EξEη havasarowyownic bxowm ξ -i ankaxowyown, ee ξ -n η-n yowraqanyowrn ndownowm en erkow areq: η -i

510. Dicowq ξ patahakan me owyownn owni cowcayin baxowm λ parametrov, isk φ-n havasaraa baxva [0, 2π] mijakay{ qowm: Gtnel η = sin(ξ + φ) patahakan me owyan ma. spasowm, ee ξ -n φ-n ankax en: 511. Dicowq ξ1 ξ2 - (a, σ2 ) parametrerov ankax, normal baxva patahakan me owyownner en: Apacowcel, or σ E max(ξ1 , ξ2 ) = a + √ , π

σ E min(ξ1 , ξ2 ) = a − √ : π

512. Dicowq (ξ, η) vektorn owni normal baxowm, nd orowm Eξ = Eη = 0, Eξ 2 = Eη 2 = 1, Eξη = ρ: Apacowcel, or √ 1−ρ : E max(ξ, η) = π 513. Dicowq ξ1 , ξ2 , ..., ξn - verjavor ma. spasowmner owneco patahakan me owyownner en: Apacowcel, or E max{ξ1 , ξ2 , ..., ξn } ≥ max{Eξ1 , Eξ2 , ..., Eξn },

E min{ξ1 , ξ2 , ..., ξn } ≤ min{Eξ1 , Eξ2 , ..., Eξn } :

514. ξ η patahakan me owyownnern ownen verjavor disper{ sianer` Dξ = σ12 , Dη = σ22 : Gtnel D(ξ+η)-i o oxman mijakayq: 515. Apacowcel, or ee ξ en, apa

η patahakan me owyownnern ankax

Dξη = DξDη + (Eξ)2 Dη + (Eη)2 Dξ,

aysinqn

Dξη ≥ DξDη :

516. Gtnel hamapatasxanabar [0, 1] [1, 3] mijakayqerowm havasaraa baxva ξ η ankax patahakan me owyownneri ξ · η artadryali ma. spasowm: 517. (ξ, η) patahakan ket havasaraa baxva R = [0, 1] × [0, 1] qa akowsow nersowm: Gtnel ζ = ξ · η patahakan me owyan ma. spasowm dispersian:

518. (ξ, η) patahakan ket havasaraa baxva (0, 0) kentron r = 1 a avi owneco rjani nersowm: Gtnel ζ = ξ · η patahakan me owyan ma. spasowm dispersian: 519. ξ patahakan me owyown havasaraa baxva [a, b] mijakayqowm: Gtnel a b-n, ee Eξ 2 = 1, Eξ = −Eξ 3 : 520. Dicowq ξ1 , ξ2 , ..., ξn patahakan me owyownnern ankax en, ownen 0-in havasar ma. spasowmner verjavor errord kargi momentner: Apacowcel, or (

n ∑ k=1

ξk

n ∑

Eξk3 :

k=1

521. Havel λ parametrov cowcayin baxowm owneco tahakan me owyan skzbnakan momentner:

pa{

522. Havel (a, σ2 ) parametrerov normal baxva ξ pata{ hakan me owyan kentronakan momentner:

523. Dicowq ξ patahakan me owyownn ndownowm verjavor vov o bacasakan x1 , x2 , ..., xn areqner: Apacowcel, or Eξ n+1 = max xi , n→∞ Eξ n 1≤i≤k lim

lim

n→∞

√ n Eξ n = max xi : 1≤i≤k

524. Apacowcel, or ee Eξ 2 = Eξ 3 = Eξ 4 , apa ξ patahakan me owyown diskret karo ndownel miayn erkow` 0 1 areqner: 525. Apacowcel, or ee Eξ 2n , Eξ 2n+1 , Eξ 2n+2 ver handisa{ nowm en vabanakan progresiayi hajordakan andamner, apa nranq irar havasar en, isk ξ me owyown diskret karo ndownel miayn 0 1 areqner: 526. ξ patahakan me owyownn ndownowm ±1, ±2 areqner yowraqanyowr 1/4 havanakanowyamb, isk η = ξ 2 : a) Gtnel ξ -i η -i hamate baxowm: b) Apacowcel, or ρ(ξ, η) = 0: g) Apacowcel, or ξ -n η-n ankax en: 527. (ξ, η) patahakan vektorn owni het yal baxman renq` η\ξ 0 1/12 1/12 1/12 0 1/4 1/4 1/8 0 1/8 Gtnel nra dispersion matric: 528. Berel erkow patahakan me owyownneri rinak, oronc ko{ relyaciayi gor akic havasar 0-i, sakayn dranq ankax en: 529. ξ

η patahakan me owyownnern ankax en P (ξ = 1) = P (ξ = −1) = 1/2, P (η = 1) = P (η = −1) = 1/4, P (η = 0) = 1/2: Kline?n ardyoq ξ · η η patahakan me owyownner a) ankax, b)

korelacva :

530. (ξ, η) patahakan vektor havasaraa baxva (0, 0), (0, 1), (1, 0) gaganer owneco e ankyan nersowm: Gtnel ξ -i η-i korelyaciayi gor akic: 531. Dicowq ξ ∼ N (0, σ): Gtnel dispersion matric:

(ξ, ξ 3 )

patahakan vektori

532. ξ1 , ξ2 , ..., ξn patahakan me owyownneric yowraqanyowr er{ kowsi korelyaciayi gor akic havasar ρ-i: Apacowcel, or ρ≥

−1 : n−1

533. ξ1 , ξ2 , ..., ξn+m , (n ≥ m) patahakan me owyownnern an{ kax en, miatesak baxva ownen verjavor dispersianer: Gtnel η1 = ξ1 + ξ2 + ... + ξn η2 = ξm+1 + ξm+2 + ... + ξm+n patahakan me owyownneri korelyaciayi gor akic: 534. Dicowq ξ η patahakan me owyownner normal en bax{ va , nd orowm Eξ = Eη = 0 dranc korelyaciayi gor akic havasar ρ-i: Gtnel ξ 2 η2 patahakan me owyownneri kore{ lyaciayi gor akic: 535. Dicowq Eξ = Eη = 0, Dξ = Dη = 1 paymannerin bavara{ ro ξ η patahakan me owyownneri korelyaciayi gor akic havasar ρ-i: Apacowcel, or E max{ξ 2 , η 2 } ≤ 1 +

536. Dicowq f (x, y) =

{

y 2 −x2 −x 8 e ,

√

1 − ρ2 :

0 < y < ∞,

−y ≤ x ≤ y

mnaca depqerowm

erka patahakan me owyan hamate xtowyan fownk{ cian : Cowyc tal, or E(ξ/η = y) = 0: (ξ, η)

537. Dicowq {

−x y e−y

f (x, y) = 0,

0 < x < ∞,

0<y<∞

mnaca depqerowm

erka patahakan me owyan hamate xtowyan fownk{ cian : Cowyc tal, or E(ξ/η = y) = y : (ξ, η)

538. Dicowq

{

f (x, y) =

e−y y ,

0 < x < y,

0<y<∞

mnaca depqerowm

erka patahakan me owyan hamate xtowyan fownk{ cian : Gtnel E(ξ 2 /η = y): (ξ, η)

539. Dicowq ξ η-n hamapatasxanabar λ1 λ2 paramet{ rerov Powasoni baxowm owneco ankax patahakan me owyown{ ner en: Gtnel E(ξ/ξ + η = n): 540. Dicowq

{

f (x, y) =

6xy(2 − x − y), 0,

0 < x < 1,

0<y<1

mnaca depqerowm

erka patahakan me owyan hamate xtowyan fownk{ cian : Gtnel E(ξ/η = y), orte 0 < y < 1: (ξ, η)

541. Dicowq f (x, y) =

{

4y(x − y)e−(x+y) , 0,

0 < x < ∞,

0≤y≤x

mnaca depqerowm

erka patahakan me owyan hamate xtowyan fownk{ cian : Gtnel E(ξ/η = y): (ξ, η)

542. Dicowq

{

f (x, y) =

−xy , 2 ye

0 < x < ∞,

0<y<2

mnaca depqerowm

erka patahakan me owyan hamate xtowyan fownk{ cian : Gtnel E(eξ/2 /η = 1): (ξ, η)

543. Dicowq

{

f (x, y) =

2e−2x x ,

0 ≤ x < ∞,

0≤y≤x

mnaca depqerowm

erka patahakan me owyan hamate xtowyan fownk{ cian : Gtnel cov(ξ, η) : (ξ, η)

544. Dicowq trva (ξ, η) erka patahakan me owyan ha{ mate xtowyan fownkcian {

f (x, y) =

1 −(y+x/y) , ye

x > 0,

y>0

mnaca depqerowm :

Gtnel Eξ , Eη, cov(ξ, η): 545. Za netowm en 3 angam: Dicowq ξ -n <1>-eri i hayt galow ivn , isk η-n` <2>-eri: Gtnel cov(ξ, η) : 546. Jravazanowm ka 100 owk, oroncic 30- karp en: Gtnel patahakanoren vercva 20 kneri mej karperi vi ma. spa{ sowm dispersian: 547. Dicowq ξ1 , ξ2 , . . . , ξn ankax miatesak baxva pataha{ kan me owyownnern ownen havasaraa baxowm [0, 1] mijakay{ qowm: Gtnel E(min(ξ1 , ξ2 , . . . , ξn )) E(max(ξ1 , ξ2 , . . . , ξn )): 548. N mard irenc glxarkner dnowm en senyakowm: Glxarkner xa nvowm en yowraqanyowr patahakanoren vercnowm mek:

Gtnel ayn mardkanc vi ma. spasowm ntrel en irenc se akan glxarkner:

dispersian, oronq

549. Enadrenq arkowm ka 2N qart, oroncic erkowsi vra nva 1, erkowsi vra nva 2 ayln: Patahakanoren vercnowm en m qart: Havel ayn zowyg qarteri vi ma. spasowm, oronq de s mnowm en arki mej: 550. Dicowq ξ1 , ξ2 , . . . , ξn - σ2 dispersiayov ankax, miatesak baxva patahakan me owyownner en: Apacowcel, or cov(ξi − ξ, ξ) = 0, orte ξ =

1∑ ξi : n n

i=1

551. Dicowq ξ -n N (0, 1) standart normal baxva pataha{ kan me owyown η = a + bξ + cξ 2 : Cowyc tal, or b ρ(ξ, η) = √ : b + 2c2

Me veri renq ebi i anhavasarowyown

a) Ee ξ ≥ 0 verjavor Eξ maematikakan spasowm owneco patahakan me owyown , apa P (ξ ≥ ε) ≤

Eξ , ε

ε>0:

b) Ee ξ patahakan me owyownn owni verjavor Dξ dispersia, apa P (|ξ − Eξ| ≥ ε) ≤

Dξ , ε2

ε>0:

Asowm en, or {ξn }n=1,2,... hajordakanowyown enarkvowm me

veri renqin, ee ξ1 , ξ2 , . . . , ξn , . . . patahakan me owyownneri

hajordakanowyan hamar tei owni (

lim P

n→∞

1∑ 1∑ ξk − Eξk < ε n n n

k=1

) =1:

k=1

ebi i eorem: Ankax sahmana ak dispersianer owneco patahakan me owyownneri {ξn }n=1,2,... hajordakanowyown en{ arkvowm me veri renqin: ebi i eoremi het anqner:

a) Be nowlii eorem`

( m ) − p < ε = 1, n→∞ n orte ε > 0 cankaca iv , m- n ankax oreri nacqowm A patahowyi handes galow ivn , p-n A patahowyi havana{ lim P

kanowyownn yowraqanyowr orowm: b) (

1∑ ξk − a < ε n n

lim P

n→∞

) = 1,

k=1

orte ε > 0 cankaca iv , ξ1 , ξ2 , . . . , ξn , . . . miatesak bax{ va verjavor dispersia owneco patahakan me owyownner en, Eξk = a (n = 1, 2, . . .)

g) Powasoni eorem`

lim P

n→∞

(

m 1∑ − pk < ε n n n

) = 1,

k=1

orte ε > 0, pk -n (k = 1, 2, . . .) A patahowyi havanakanowyownn k-rd orowm: m- n ankax orerowm A patahowyi er owmneri ivn :

Markovi eorem: {ξn }n=1,2,... hajordakanowyown enarkvowm me veri renqin, ee ) ( n ∑ lim ξk = 0 : D n→∞ n2 k=1

Kasenq, or ηn patahakan me owyownneri hajordakanowyown st baxman zowgamitowm η patahakan me owyan, ee nranc Fn (x) baxman fownkcianer zowgamitowm en F (x) baxman fownk{ ciayin F (x) downkciayi bolor anndhatowyan keterowm: Kar D growm en` ηn −→ η: Kasenq, or ηn patahakan me owyownneri hajordakanowyow{ n st havanakanowyan zowgamitowm η patahakan me owya{ n, ee kamayakan ε > 0 vi hamar P {ω : |ηn (ω) − η(ω)| ≥ ε} → 0,

erb

n→∞:

P Kar growm en` ηn −→ η: Kasenq, or ηn patahakan me owyownneri hajordakanowyow{ n hamarya havasti zowgamitowm η patahakan me owyan, ee −−→ η(ω)} = 1 : P {ω : η(ω) − n→∞

h.h. Kar growm en` ηn −→ η: Kasenq, or ηn patahakan me owyownneri hajordakanowyow{ n r-rd kargi mijin imastov (r ∈ (0, +∞)) zowgamitowm η pa{ tahakan me owyan, ee E|ηn |r < ∞ bolor n-eri hamar

E|ηn − η|r → 0,

erb

n→∞:

Kar growm en` ηn −→ η: (r)

{ξn }n=1,2,... patahakan me owyownneri hajordakanowyown (Eξk = 0, k = 1, 2, ...) enarkvowm me veri oweacva renqin, ee (

1∑ ξk → 0 n n

k=1

)

=1:

A 552. gtvelov ebi i anhavasarowyownic, gnahatel hete{ vyal havanakanowyown` P (|ξ − Eξ| ≥ 2σ), orte σ2 = Dξ : 553. gtvelov ebi i anhavasarowyownic, gnahatel hete{ vyal havanakanowyown` P (|ξ − Eξ| < 3σ), orte σ2 = Dξ : 554. Trva ξ patahakan me owyan havanakanowyownneri baxman renq` ξ 0,3 0,6 : P 0,2 0,8 Gnahatel het yal havanakanowyown` P (|ξ − Eξ| < 0, 2): 555. Trva ξ patahakan me owyan havanakanowyownneri baxman renq` ξ 0,1 0,4 0.6 : P 0,2 0,3 0,5 √ Gnahatel het yal havanakanowyown` P (|ξ − Eξ| < 0, 4): 556. Trva ξ patahakan me owyan havanakanowyownneri baxman renq` ξ P

-3 -2 -1 0 1 2 3 0,1 0,1 0,2 0,2 0,2 0,1 0,1

Gtnel P (|ξ| ≥ 1) havanakanowyan grit areq ayn hamema{ tel ebi i anhavasarowyownic stacva gnahatakani het: 557. Mek townk kartofili mijin qa 100 g : Gnahatel pa{ tahakan townk kartofili ki 300 gramic o aveli linelow ha{ vanakanowyown: 558. Me vov k owmneri depqowm marmni mijin ki stacvel 5,25 kg: K i mijin qa akowsayin eowm havasar 0,02 kg:

Gnahatel mek patahakan k man ardyownqi 5,2 kg-ic 5,3 kg-i sahmannerowm gtnvelow havanakanowyown: 559. Tvyal teamasi mek tarva nacqowm teowmneri qana{ ki maematikakan spasowm 55 sm : Gnahatel ayd teamasowm 155 sm-ic o pakas qanakowyamb teowmner linelow havanaka{ nowyown: 560. Dicowq ξ patahakan me owyan ma. spasowm disper{ sian havasar en 20-i: Gnahatel P (0 ≤ ξ ≤ 40): 561. gtvelov ebi i anhavasarowyownic, gtnel drami 100 netowmneri depqowm gerbi er owmneri haaxowyan mek orowm i hayt galow havanakanowyownic owneca eman 0,1-in gera{ zancelow havanakanowyown: 562. Oroel | mn − 12 | < 100 patahowyi havanakanowyan sto{ rin ezr, orte m- n ankax orerowm A patahowyi er owmne{ ri ivn A patahowyi havanakanowyown ankax oreric yowraqanyowrowm havasar 12 , ee n = 10000; n = 100000:

563. A patahowyi havanakanowyown ankax oreric yow{ raqanyowrowm havasar 0,25: gtvelov ebi i anhavasarow{ yownic, gnahatel het yal havanakanowyown` P (150 ≤ m ≤ 250), orte m- 800 ankax orerowm A patahowyi er owmneri ivn : 564. gtvelov ebi i anhavasarowyownic, gnahatel het { yal havanakanowyown` P (40 ≤ m ≤ 60), orte m- 100 ankax

orerowm A patahowyi er owmneri ivn : A patahowyi ha{ vanakanowyown ankax oreric yowraqanyowrowm havasar 0,5-i: 565. 0,99-ic o pakas havanakanowyamb, inpisi verin ezr kareli nel | mn − 13 | artahaytowyan hamar, orte m- n ankax

orerowm A patahowyi er owmneri ivn A patahowyi ha{ vanakanowyown ankax oreric yowraqanyowrowm havasar 13 , ee n = 10000: 566. gtvelov ebi i anhavasarowyownic gtnel ε-, ee P (|ξ − Eξ| < ε) > 0, 9 Dξ = 0, 009: 567. Owsano hannowm qnnowyown, ori ardyownq 75 mijinov 25 dispersiayov patahakan me owyown : a) Gtnel qnnowyown 85 miavoric barr stanalow havanakanowyan verin areq: b) Gnahatel owsanoi staca miavori 65-ic 85 miavori mij linelow havanakanowyown: g) Qani? owsano petq hanni qn{ nowyown, orpeszi 0,9-ic o pakas havanakanowyamb nranc sta{ ca miavori mijin eowm 75 areqic lini 5-ic o aveli: 568. Dicowq ξ -n (0, 10) mijakayqowm havasaraa baxva

patahakan me owyown : Gnahatel P (|ξ −5| > 4) havanakanow{ yown hamematel ayn grit areqi het: 569. Trva ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` ξn pn

√ − 3

√

(n = 1, 2, . . .) :

Apacowcel, or ayd hajordakanowyown enarkvowm me veri renqin: 570. Trva ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` ξn pn

√ − 2

√

(n = 1, 2, . . .) :

Apacowcel, or ayd hajordakanowyown enarkvowm me veri renqin: 571. Trva ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` ξn −nα nα (n = 1, 2, . . .) : pn

1−

2n−1

Apacowcel, or ayd hajordakanowyown enarkvowm me veri renqin: 572. Trva ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` ξn pn

−α

n 2n+1

n+1 2n+1

(n = 1, 2, . . .) :

Apacowcel, or ayd hajordakanowyown enarkvowm me veri renqin: 573. Trva ξ2 , ξ3 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` √ √ ξn − n n pn

√1 n

1−

√2 n

√1 n

(n = 2, 3, . . .) :

Ayd hajordakanowyown enarkvow?m me veri renqin, e o (gtvel Markovi eoremic): 574. Trva ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` ξn −nα nα (n = 1, 2, . . .) : pn

2n2

1−

2n2

Ayd hajordakanowyan nkatmamb kareli ? kira el ebi i eo{ rem: 575. Trva ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` ξn pn

√ − ln n

√ ln n

(n = 1, 2, . . .) :

Ayd hajordakanowyown enarkvowm ? me veri renqin, e` o (gtvel Markovi eoremic):

576. Apacowcel, or ee ξ patahakan me owyown owni verja{ vor orrord kargi µ4 kentronakan moment, apa P (|ξ − Eξ| ≥ ε) ≤

µ4 : ε4

577. Dicowq f (x) > 0 nvazo fownkcia : Apacowcel, or ee goyowyown owni Ef (|ξ − Eξ|) ma. spasowm, apa P (|ξ − Eξ| ≥ ε) ≤

Ef (|ξ − Eξ|) : f (ε)

578. Apacowcel, or cankaca ξ patahakan me owyan hamar tei owni het yal anhavasarowyown P (ξ > t2 + ln(Eeξ )) < e−t :

579. Oroel Be nowlii modelowm ankax oreri n iv, ori dep{ qowm ( ) P

m 1 − < n

> 0, 99 :

580. Za netelis xaaco ahowm 8$, ee bacvowm <6>- varowm 1$` haka ak depqowm: Gnahatel za i n = 1000 netowm{ neri depqowm xaacoi aha gowmar 250$-ic me linelow hava{ nakanowyown:

581. Za netelis xaaco ahowm 4 $, ee bacvowm <6>- varowm 1$` haka ak depqowm: Gnahatel za i n = 10000 netowm{ neri depqowm xaacoi partva gowmar 200$-ic o pakas line{ low havanakanowyown: 582. Erkow za netelis xaaco ahowm aynqan dram, orqan za eri vra bacva veri tarberowyownn , ee dranq mimyanc havasar en, haka ak depqowm xaaco varowm aynqan dram, orqan za eri vra bacva veri gowmarn : 0,99 havanakanow{ yamb inpisi ahowm kareli eraxavorel xaacoin, ee xa krknvowm n = 3000 angam: 583. Apacowcel 

P

n ∑

v  u n u∑ pi − m ≥ 2tt pi qi  < e−t ,

i=1

anhavasarowyown, orte pi -n i -rd orowm A patahowyi ha{ vanakanowyownn (qi = 1 − pi ), m-` n ankax orerowm A pa{ tahowyi handes galow ivn : 584. 2500 ankax patahakan me owyownneric yowraqanyow{ ri dispersian i gerazancowm 5-in: Gnahatel ayd pataha{ kan me owyownneri mijin vabanakani` nranc maematikakan spasowmneri mijin vabanakanic owneca eman 0,4-in gera{ zancelow havanakanowyown: 585. Trva ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ neri hajordakanowyown, nd orowm yowraqanyowr ξn owni het yal baxman renq` ξn pn s-i

n−s

(n = 1, 2, . . .) :

?or areqi depqowm hajordakanowyown kenarkvi me veri renqin (gtvel Markovi eoremic):

586. Ankax oreric yowraqanyowrowm A patahowyi handes galow havanakanowyown` P (A) = p, (P (Ā) = 1 − p): Nanakenq ξi -ov (i = 1, 2, . . .) i-rd orowm A patahowyi handes galow iv: Apacowcel, or ayd patahakan me owyownneri hajordakanowyow{ n enarkvowm me veri renqin (gtvel Markovi eoremic): 587. Apacowcel, or ee patahakan me owyownneri disper{ sianeri hajordakanowyown sahmana ak , nd orowm yowraqan{ yowr patahakan me owyown kaxva miayn har an pataha{ kan me owyownneric, enarkvowm me veri renqin: 588. Trva ξ1 , ξ2 , . . . , ξn , . . . patahakan me owyownneri ha{ jordakanowyown, nd orowm Dξn ≤ C(n = 1, 2, . . .) rij → 0, erb |i−j| → ∞ ( rij -n ξi ξj patahakan me owyownneri korelyaciayi gor akicn ): Apacowcel, or ayd hajordakanowyown enarkvowm me veri renqin (Be teyni eorem): 589. Apacowcel, or Dξnn → 0, erb n → ∞ paymanin bavararo ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyownneri hajordaka{ nowyown enarkvowm me veri renqin (Xinini eorem): p 590. Apacowcel, or ee |ξn | ≤ k n→∞ lim ξn = a, apa lim Eξn = n→∞ a: Cowyc tal, or |ξn | ≤ k (k = 1, 2, . . .) payman kar or : Inpe?s kareli owlacnel ayd payman:

Bnowagri fownkcia ξ

patahakan me owyan bnowagri fownkcia kovowm ∫∞ φξ (t) = Ee

itξ

eitx dFξ (x)

= −∞

fownkcian, orte Fξ (x)- ξ -i baxman fownkcian : Diskret pa{ tahakan me owyan hamar bnowagri fownkcian klini` φξ (t) =

∑

eitxk pk ,

orte pk = P (ξ

owyan hamar`

= xk ):

Bacarak anndhat patahakan me{ ∫∞ eitx fξ (x)dx,

φξ (t) = −∞

orte fξ (x)- ξ patahakan me owyan xtowyan fownkcian : Bnowagri fownkcian havasaraa anndhat t-i nkatmamb amboj a ancqi vra, nd orowm φ(0) = 1,

|φ(t)| ≤ 1,

−∞ < t < ∞ :

Ee η = aξ + b, apa φη (t) = φξ (at)eitb :

Ee ξ

η -n

ankax patahakan me owyownner en, apa φξ+η (t) = φξ (t) · φη (t) :

Ee ξ patahakan me owyan n-rd kargi skzbnakan moment verjavor , apa bnowagri fownkcian n angam diferenceli φ(k) (0) = ik · Eξ k ,

k≤n:

Ee φξ (t) bnowagri fownkcian bacarak integreli amboj a ancqi vra, apa fξ (x) xtowyan fownkcian artahaytvowm φξ (t) bnowagri fownkciayi mijocov het yal bana ov` fξ (x) = 2π

∫∞

e−itx φξ (t)dt :

−∞

ndhanowr depqowm it het yal rjman bana ` Fξ (x2 ) − Fξ (x1 ) = lim 2π c→∞

∫c

−c

orte isk x1

eitx1 − eitx2 φξ (t)dt, it

Fξ (x)- ξ patahakan x2 - Fξ (x) fownkciayi

me owyan baxman fownkcian , anndhatowyan keter en: Owi sahmanayin eorem. Ee Fn (x) (n = 1, 2, . . .) baxman fownkcianeri hajordakanowyown zowgamitowm F (x) baxman fownkciayin, erb n → ∞ verjini anndhatowyan keterowm, apa hamapatasxan φn (t) (n = 1, 2, . . .) bnowagri fownkcianeri hajordakanowyown, havasaraa st t-i, yowraqanyowr ver{ javor mijakayqowm zowgamitowm , erb n → ∞, F (x) baxman fownkciayin hamapatasxano φ(t) bnowagri fownkciayin: Hakadar sahmanayin eorem. Dicowq φn (t) (n = 1, 2, . . .) bnowagri fownkcianeri hajordakanowyown t = 0 ketowm zowgami{ towm φ(t) anndhat fownkciayin, erb n → ∞: Ayd depqowm ha{ mapatasxan Fn (x) (n = 1, 2, . . .) baxman fownkcianeri hajor{ dakanowyown, erb n → ∞, zowgamitowm φ(t)-in hamapatas{ xano F (x) baxman fownkciayin nra anndhatowyan keterowm: Sahmanowm. Kasenq, or φ(t) kompleqs areqani fownkcian o bacasakanoren orova , ee cankaca n ∈ N , ∀ t1 , t2 , ..., tn ∈ R, ∀λ1 , λ2 , ..., λn ∈ C hamar n n ∑ ∑ i=1 j=1

φ(ti − tj )λi λj ≥ 0,

orte λj -n λj -i hamalow n : Boxneri eorem. Dicowq φ(t)-n anndhat fownkcia φ(0) = 1: Orpeszi φ(t)-n lini bnowagri fownkcia anhraet bavarar, or na lini o bacasakanoren orova : Het anq. Ee φk (t) bnowagri fownkcianer en, k = 1, 2, ..., apa ∞ ∑ φ(t) = ck φk (t) s bnowagri fownkcia , orte ck ≥ 0 ∞ ∑ k=1

k=1

ck = 1:

Kentronakan sahmanayin eorem eorem. Ee ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyownneri hajordakanowyown cankaca τ > 0 vi hamar bavararowm ∫

n 1 ∑ lim 2 n→∞ Bn

(x − ak )2 dFk (x) = 0,

k=1 |x−a |>τ B n k

paymanin (Lindeberg), orte Bn2

ak = Eξk ,

n ∑

Dξk ,

k = 1, 2, . . . ,

n = 1, 2, . . . ,

k=1

Fk (x) = P (ξk < x),

apa

( lim P

n→∞

n 1 ∑ (ξk − ak ) < x Bn k=1

)

=√ 2π

∫x

e− 2 dz

−∞

havasaraa st x-i:

Lyapownovi eorem. Ee ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan

me owyownneri hajordakanowyan hamar goyowyown owni iv, aynpisin or lim

n→∞

n ∑

Bn2+δ

k=1

E|ξk − ak |2+δ = 0,

δ > 0

apa it (*)-: Het anq. Ee ξ1 , ξ2 , . . . , ξn , . . . ankax patahakan me owyown{ ner miatesak en baxva Dξn ̸= 0, apa it (*)-:

A 591. ξ patahakan me owyownn owni havanakanowyownneri bax{ man renq` ξ -1 1 :

Gtnel ξ -i bnowagri fownkcian: 592. ξ patahakan me owyownn owni havanakanowyownneri bax{ man renq` ξ -2 0 2 : p

Gtnel ξ -i bnowagri fownkcian: 593. Gtnel n ankax orerowm A patahowyi handes galow vi bnowagri fownkcian, ee A patahowyi k-rd orowm han{ des galow havanakanowyown` P (A) = pk k = 1, 2, . . . , n: 594. ξ patahakan me owyownn owni het yal baxman fownk{ cian`   0, F (x) = p,   1,

(0 < p < 1):

x≤a a<x≤b x>b

Gtnel ξ -i bnowagri fownkcian:

595. Gtnel [−a, a] mijakayqowm havasaraa baxva pa{ tahakan me owyan bnowagri fownkcian:

596. ξ patahakan me owyownn owni het yal xtowyan fownk{ cian` { f (x) =

a−|x| , a2

|x| ≥ a : |x| ≤ a

Gtnel ξ -i bnowagri fownkcian: 597. Gtnel f (x) =

a −a|x| e ,

a>0

0, e−x ,

x≤0 x>0

Laplasi baxman bnowagri fownkcian: 598. Gtnel { f (x) =

xtowyown owneco cowcayin baxman bnowagri fownkcian skzbnakan momentner: 599. Gtnel [a, b] mijakayqowm havasaraa baxva pata{ hakan me owyan bnowagri fownkcian skzbnakan momentner: 600. Dicowq trva ξ patahakan me owyan bnowagri fownk{ cian` φξ (t) = e3(eit −1) : Gtnel P (ξ = 0) havanakanowyown: 601. Dicowq trva ξ patahakan me owyan φξ (t) bnowagri fownkcian: Gtnel ξ patahakan me owyan baxowm, ee a) φξ (t) = cos t b) φξ (t) = cos2 t:

602. Gtnel [a, b] mijakayqowm Simpsoni baxman (e ankyown baxman) bnowagri fownkcian: 603. Gtnel f (x) =

π(1 + x2 )

xtowyamb Koow baxman bnowagri fownkcian: 604. Gtnel binomakan baxman bnowagri fownkcian mijocov havel maematikakan spasowm dispersian:

dra

605. Gtnel P (ξ = m) =

am , (1 + a)m+1

a > 0,

m = 0, 1, 2, ...

Paskali baxman bnowagri fownkcian nra mijocov havel maematikakan spasowm dispersian: 606. ξ patahakan me owyownn owni 0 σ2 parametrerov nor{ mal baxowm: Bnowagri fownkciayi gnowyamb gtnel nra bolor kentronakan momentner: 607. Gtnel f (x) =

{

x<0 x ≥ 0, a > 0, λ > 0

aλ λ−1 −ax e , Γ(λ) x

xtowyan fownkcia owneco patahakan me owyan bnowagri fownkcian skzbnakan momentner: 608. Apacowcel Powasoni normal baxowmneri kayownowyown: 609. ξ patahakan me owyan bnowagri fownkcian ` φ(t) = e−a|t| ,

a>0:

Gtnel ayd patahakan me owyan xtowyan fownkcian: 610. ξ patahakan me owyan bnowagri fownkcian ` φ(t) =

: 1 + t2

Gtnel ayd patahakan me owyan xtowyan fownkcian: 611. Apacowcel, or φ1 (t) =

∞ ∑

ak cos kt

k=1

φ2 (t) =

∞ ∑

ak eiλk t

k=1

∞ ∑

fownkcianer, orte ak ≥ 0, ak = 1, bnowagri fownkcianer k=1 en oroel hamapatasxan baxman fownkcianer: 612. Apacowcel, or 1. e−i|t| 2. 1−i|t| {

3. φ(t) =

0, 1 − t2 ,

|t| > 1 |t| ≤ 1

4. e−t2 (π−arctan t) fownkcianer bnowagri fownkcianer en: 613. Apacowcel het yal` orpeszi patahakan me owyown li{ ni hamaa skzbnaketi nkatmamb` anhraet bavarar, or bnowagri fownkcian ndowni miayn irakan areqner: 614. Apacowcel, or cankaca irakan φ(t) bnowagri fownk{ cia bavararowm het yal anhavasarowyan` 1 − φ(2t) ≤ 4(1 − φ(t))

cankaca t ∈ R-i hamar: n`

615. Apacowcel φ(t) bnowagri fownkciayi het yal hatkowyow{ |φ(t + h) − φ(t)| ≤

√

2[1 − Reφ(h)]

(Rez - z -i irakan masn ): 616. Apacowcel, or ee φ(t) bnowagri fownkcian aynpisin , or φ(h) = 1 (h ̸= 0), apa φ(t)-n h parberowyamb parberakan fownkcia :

617. Apacowcel, or ee φ(t)-n bnowagri fownkcia , apa ψ(t) = t

∫t φ(z)dz

fownkcian nowynpes klini bnowagri fownkcia: 618. Dicowq ξ η-n binomakan baxowm owneco ankax pa{ tahakan me owyownner en hamapatasxanabar (n, p) (m, p) parametrerov: I?n baxowm kownena ξ + η patahakan me ow{ yown: 619. Dicowq ξ η-n Powasoni baxowm owneco ankax pata{ hakan me owyownner en hamapatasxanabar λ1 λ2 paramet{ rerov: I?n baxowm kownena ξ + η patahakan me owyown: 620. Apacowcel, or ee φ(t)-n bnowagri fownkcia , apa Reφ(t) s bnowagri fownkcia : 621. Apacowcel, or ee φ(t)-n bnowagri fownkcia , apa s bnowagri fownkcia :

ψ(t) = eφ(t)−1

622. Apacowcel, or ee φ(t)-n bnowagri fownkcia , apa s bnowagri fownkcia :

ψ(t) = |φ(t)|2

623. Apacowcel, or ee φ(t)-n bnowagri fownkcia , apa − 1 s bnowagri fownkcia : ψ(t) = 2−φ(t) 624. Apacowcel, or φ(t) = 13 e−t

2 /2

+ 23 e7(e

it −1)

bnowagri fownkcia : 625. Dicowq trva ξ patahakan me owyan φξ (t) bnowagri fownkcian: Gtnel ξ patahakan me owyan xtowyan fownkcian, ee φ(t) = e−t2 :

626. Dicowq trva ξ patahakan me owyan φξ (t) =

sin t t

bnowagri fownkcian: Gtnel ξ patahakan me owyan xtowyan fownkcian: 627. Dicowq trva ξ patahakan me owyan φξ (t) = e−|t| cos t

bnowagri fownkcian: Gtnel ξ patahakan me owyan xtowyan fownkcian: 628. Dicowq trva ξ patahakan me owyan φξ (t) =

1 − it

bnowagri fownkcian: Gtnel ξ patahakan me owyan xtowyan fownkcian: 629. Dicowq φ(t)-n bnowagri fownkcia : Apacowcel, or can{ kaca t ∈ R hamar it a) 1 − |φ(2t)|2 ≤ 4(1 − |φ(t)|) b) 1 − Reφ(2t) ≤ 2(1 − (Reφ(t))2 ) g) 1 − |φ(2t)| ≤ 2(1 − |φ(t)|2 ) d) 1 − |φ(2t)| ≤ 4(1 − |φ(t)|) : 630. Low el 567 xndri g) ket` gtvelov kentronakan sahmanayin eoremic: 631. ξ patahakan me owyownn owni het yal xtowyan fownkcian` { f (x) =

0, aλ λ−1 −ax e , Γ(λ) x

x≤0 x > 0,

a > 0,

λ>0:

√ Apacowcel, or erb λ → ∞, aξ−λ patahakan me owyown st λ baxman gtowm a = 0, σ = 1 parametrerov normal baxman:

632. ξ patahakan me owyownn owni Powasoni baxowm λ pa{ √ patahakan me ow{ rametrov: Apacowcel, or λ → ∞ depqowm ξ−λ λ yown st baxman gtowm a = 0, σ = 1 parametrerov normal baxman: 633. 4500 ankax, miatesak baxva patahakan me owyown{ neric yowraqanyowri dispersian havasar 5-i: Gtnel ayd pa{ tahakan me owyownneri mijin vabanakani` nranc maemati{ kakan spasowmic owneca eman 0,04-in gerazancelow havana{ kanowyown: 634. η patahakan me owyown irenic nerkayacnowm 10000 ankax miatesak baxva σ = 2 mijin qa akowsayin eowm owneco patahakan me owyownneri mijin vabanakan: 0,9544 havanakanowyamb i?n me agowyn eowm kareli aknkalel η pa{ tahakan me owyan nra maematikakan spasman hamar: 635. η patahakan me owyown irenic nerkayacnowm ankax miatesak baxva , σ2 = 5 dispersia owneco patahakan me{

owyownneri mijin vabanakan: Orqa?n petq lini ayd pata{ hakan me owyownneri iv, orpeszi 0,9973 havanakanowyamb η patahakan me owyan nra maematikakan spasman eowm gerazanci 0,01 areq: 636. η patahakan me owyown irenic nerkayacnowm 3200 an{ kax miatesak baxva , Eξ = 3 mijinov Dξ = 2 dispersiayov patahakan me owyownneri mijin vabanakan: Gnahatel P (2, 95 ≤ η ≤ 3) havanakanowyown: 637. Tei ow?nen ardyoq me veri renq kentronakan sahma{ nayin eorem a) P (ξk = ±2k ) = 12 ,

b) P (ξk = ±2k ) = 2−(2k+1) , P (ξk = 0) = 1 − 2−2k , g) P (ξk = ±k) = 12 k− 2 , P (ξk = 0) = 1 − k− 2 , (k ≥ 1) baxman renqner owneco ankax patahakan me owyownneri hamar: 638. marit ? ardyoq het yal pndowm` D

ηn −→ η

⇒

ηn − η −→ 0 :

P 639. Dicowq ξn −→ ξ

ξn −→ η :

P 640. Dicowq ξn −→ ξ

Apacowcel, or P (ξ = η) = 1:

ηn −→ η : Apacowcel, or P aξn + bηn −→ aξ + bη , orte a b-n hastatownner

a) P b) |ξn | −→ |ξ|, P g) ξn ηn −→ ξη:

P P 641. Dicowq ξn −→ ξ ηn −→ η kamayakan ε > 0 vi hamar

P (|ξn − ηn | ≥ ε) → 0, h.h. 642. Dicowq ξn −→ ξ

P (ξ = η) = 1:

erb

en,

Apacowcel, or

n→∞:

ηn −→ η : Apacowcel, or h.h. aξn + bηn −→ aξ + bη , a b-n hastatownner en, h.h.

a) h.h. b) |ξn | −→ |ξ|, h.h. g) ξn ηn −→ ξη: 643. Dicowq P ξn −→ a:

ξn −→ a, D

D 644. Dicowq ηn −→ η ξ + η:

orte a-n hastatown : Apacowcel, or ξn −→ ξ : D

D 645. Apacowcel, or ee ξn −→ ξ D a) ξn + ηn −→ ξ , P b) ξn ηn −→ 0:

D it ? ardyoq, or ξn + ηn −→

ηn −→ 0, P

apa

φ(x) = x

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

0,3989

x √1 e− 2 2π

Ayowsak 1 3984 3980 3977 3939 3932 3935 3857 3847 3836 3739 3726 3712 3580 3572 3555 3410 3391 3372 3209 3187 3166 2989 2966 2943 2756 2732 2709 2516 2492 2468

1,0 0,2420 2396 2371 2347 2323 2299 2275 2251 1,1 2179 2155 2131 2107 2083 2059 2036 2012 1,2 1942 1919 1895 1872 1849 1826 1804 1781 1,3 1714 1691 1669 1647 1626 1604 1582 1561 1,4 1497 1476 1456 1435 1415 1394 1374 1354 1,5 1295 1276 1257 1238 1219 1200 1182 1163 1,6 1109 1092 1074 1057 1040 1023 1006 0989 1,7 1940 0925 0909 0893 0878 0863 0848 0833 1,8 0790 0775 0761 0748 0734 0721 0707 0694 1,9 0656 0644 0632 0620 0608 0596 0584 0573

2,0 0,0540 0529 0519 0508 0498 0488 0478 0468 0459 0449 2,1 0440 0431 0422 0413 0404 0396 0386 0379 0371 0363 2,2 0355 0347 0339 0332 0325 0317 0310 0303 0297 0290 2,3 0283 0277 0270 0264 0256 0252 0246 0241 0235 0229 2,4 0224 0219 0213 0208 0203 0198 0194 0189 0184 0180 2,5 0175 0171 0167 0163 0158 0154 0151 0147 01443 0139 2,6 0136 0132 0129 0126 0122 0119 0116 0113 0110 0107 2,7 0104 0101 0099 0096 0093 0091 0088 0086 0084 0081 2,8 0079 0077 0075 0073 0071 0069 0067 0065 0063 0061 2,9 0060 0058 0056 0055 0053 0061 0050 0048 0047 0046

3,0 0,0044 0043 0042 0040 0039 0038 0037 0036 0035 0034 3,1 0033 0032 0031 0030 0029 0028 0027 0026 0025 0025 3,2 0024 0024 0022 0022 0021 0020 0020 0019 0018 0018 3,3 0017 0016 0016 0016 0015 0015 0014 0014 0013 0013 3,4 0012 0012 0012 0011 0011 0010 0010 0010 0009 0009 3,5 0009 0008 0008 0008 0008 0007 0007 0007 0007 0006 3,6 0006 0006 0006 0005 0005 0005 0005 0005 0005 0004 3,7 0004 0004 0004 0004 0004 0003 0003 0003 0003 0003 3,8 0003 0003 0003 0003 0003 0002 0002 0002 0002 0002 3,9 0002 0002 0002 0002 0002 0002 0002 0002 0001 0001 Φ(x) =

√1 2π

∫x

e−

Ayowsak 2

x 0 1 0,0 0,0000 0039 0079 0119 0159 0199 0239 0279 0318 0358 0,1 0398 0438 0477 0517 0556 0596 0635 0674 0714 0753 0,2 0792 0831 0870 0909 0948 0987 1025 1064 1102 1140 0,3 1179 1217 1255 1293 1330 1368 1405 1443 1480 1517 0,4 1554 1591 1627 1664 1700 1736 1772 1808 1843 1879 0,5 1914 1949 1984 2019 2054 2088 2122 2156 2190 2224 0,6 2257 2290 2323 2356 2389 2421 2453 2485 2517 2549 0,7 2580 2611 2642 2673 2703 2733 2763 2793 2823 2852 0,8 2881 2910 2938 2967 2995 3029 3051 3078 3105 3132 0,9 3159 3185 3212 3238 3263 3289 3314 3339 3364 3389

1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8

1,9 4712 4719 4725 2,0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,5 4,0 4,5 5,0

4772 4777 4821 4825 4861 4864 4892 4895 4918 4920 4937 4939 4953 4954 4965 4966 4974 4975 4981 4981 0,49865 49977 49996 49999 49999

4783 4788 4830 4834 4867 4871 4898 4901 4922 4924 4941 4943 4956 4957 4967 4968 4976 4976 4982 4983 3,1 49903 3,6 49984

4793 4798 4838 4842 4874 4877 4903 4906 4925 4928 4944 4945 4958 4859 4969 4970 4977 4978 4983 4984 3,2 49931 3,7 49989

4803 4807 4846 4850 4880 4884 4908 4911 4930 4932 4947 4949 4960 4962 4971 4972 4978 4979 4984 4985 3,3 49952 3,8 49993

4812 4816 4853 4857 4887 4889 4913 4915 4934 4936 4950 4952 4963 4964 4972 4973 4980 4980 4985 4986 3,4 49966 3,9 49995

PATASXANNER

11. x = B̄ : 19. A = ∅, B = (0, 1) r { 12 }: 27. 49, 42: 28. A6n = 151200: 29. 1680: 30. n(n−3) : 31. 8! = 40320: 32. (n − 1)!(n − 2): 33. 6912: 34. 2520: 35. Cn4 : 36. 462, 252: N −1 N −1 10 = 8008: 40. a) C n 38. CNn +n−1 , Cn−1 : 39. C16 N +n−1 , b) Cn−1 : n 41. CN +n−1 : 42. a) 720, b) 672, g) 384, d) 216, e) 576: 43. 34 : 44. a) 61 , b) 56 : 45. 0, 1512 : 46. a) ≈ 0, 0004, b) ≈ 0, 1499: k ≈ 0, 0003: 47. a) lk−1 , b) Alkl : 48. 0,4: 49. a) 49 ≈ 0, 44... , b) 3024 n C C a 50. 24 : 52. CNNn−k : 53. ≈ 0, 1788 : 54. a) 91 ≈ 0, 2637: 51. C 2 a+b

≈ 0, 5263, b) ≈ 0, 418: r ∑

n−m

55. ≈ 0, 39: 56. CM CCNNn −M : 57.

s ·C r−s Cm n−m r Cn

∑

i=3

r−1

6−i C6i ·C43 C49

≈

−3 : 59. 551 ≈ 0, 1818 . . .: 60. CCNr−1 : N −1 s=1 61. a) ≈ 0, 162, b) ≈ 0, 155: 62. a) ≈ 0, 0001, b) ≈ 0, 13 , g) ≈ 0, 1484: 63. a) ≈ 0, 504, b) ≈ 0, 432 , g) ≈ 0, 027, d) ≈ 0, 036, m n−m−2 , (0 ≤ m ≤ n − 2), e) ≈ 0, 001: 64. a) 13 , b) Cn−2 ·23n m ·2n−m Cn g) 3n , (0 ≤ m ≤ n), d) m0 !m1n!!m2 !3n , m0 + m1 + m2 = n: n−m 49−3l 65. Cn+k−m : 66. a) 270725 , b) 221 , g) 270725 : 67. a) 0, 1055, b) C45213 , n Cn+k g) (4!) 52!(13!) : 68. 67 69. 216 ≈ 0, 4213: 70. 27 91 ≈ 0, 7363: 29 ≈ 0, 931: −9 −8 71. a) 3 ≈ 0, 00005, b) 3 ≈ 0, 00015, g) 1 − ( 23 )9 ≈ 0, 974, d) (3!)9!3 ·39 ≈ 0, 0854: 72. 1212!12 ≈ 0, 00005: 73. nn!n : 74. a) 13 , b) 13 , , b) n−1 , g) (n−2)(n−1) , d) (n−2)(n−1) : g) 31 : 75. O: 76. a) n−1 77. a) n , b) n , g) n(n−1) , d) n(n−1) : 78. a) ≈ 0, 0038, b) ≈ 0, 0006, g) 16 , d) ≈ 0, 0026: 79. 0,2: 80. ≈ 0, 0055: 81. ≈ 0, 0809: 83. 0,21672: n! 84. a) 1/n, b) (n − 1)k−1 /nk : 85. 1) a) nk!k , b) (n−k)!n k , 2) a) k!(n−1)! n!(n−1)! k!(n−k)! k!(n−k)! , b) 1, 4) a) n! , n! (n+k−1)! , b) (n+k−1)!(n−k)! , 3) a) n−m−1 n−1 m Cn Cr−1 C2n−1 b) 1: 86. 2(n−r−1) : 87. : 88. 89. ≈ 4265: r n−1 : Cn+r−1 n(n−1) C3n−1 n−m+1 90. 0, 5: 91. ≈ 0, 9508: 92. n+1 , ee m ≤ n 0, ee m > n: 2n n! 93. n+m : 94. a) (2n−1)!! , b) (2n−1)!! : 95. a) 216 ≈ 0, 1991,

0, 019:

58.

≈ 0, 0926:

96. (1 − n2 )2r −2 : 97. a)  2k  N ,

(1 − n1 )r−1 ,

r−1 ∏

(1 − nk ):

k=1

l=0 1 ≤ l ≤ 8 ; lim pN = N →∞  2k+2 , l = N 1 N : 99. p = [ ] → , orte -n vi amboj masn (: 100. N N k k) ( p2 = N12 [ N2 ]2 + (N − [ N2 ])2 = 1 − N2 [ N2 ] + N22 [ N2 ]2 , p3 = N12 [ N3 ]2 + ) (N − [ N3 ])2 = 1 − N2 [ N3 ] + N22 [ N3 ]2 , p2 < p3 , ee N ≥ 3: 101. a)

98. Ee N

= 10k + l,

apa pn =  2k+1 N ,

, b) Cn (n−1) , nn g) n1 , d) n1k : 102. a) n!1 , b) A1kn , ee bolor j1 < j2 < ... < jk { tarber en 0, ee j1 < j2 < 2... < jk -i mij kan miatesakner, ! g) n1 : 103. 1) 1 − NNn! , 2) NN!CNN+1+1 , 3) ( 12 CN2 +1 CN2 −1 + CN3 +1 ) · NNN +1 : (a−2r) a−r 104. 0,0769, 0,03116: 105.√ 5 : 106. a : 107. a) , b) 1 − a 3 3 r 2 4( a ) : 108. a) π , b) 4π : 109. 2 : 110. a) 12 , b) 0: 111. a) 3 , b) 12 : 112. k(2 − k): 113. ≈ 0, 121: 114. 2) a) 1 − (1 − z) , b) z(1−ln z), g) 1−(1−z) , d) z , e) 2z , ee z ≤ 2 1−2(1−z)2 , ee z > 21 : 115. 1+38ln 2 ≈ 0, 3849: 116. ≈ 0, 21: 117. 1 − (1 − t t 2l T1 )(1 − T2 ): 118. πa : 120. a) 4 , b) 1: 121. a) 4 , b) 0: 122. (l−d−2r)(l cos φ−d−2r) l h , ee 0 ≤ arccos d+2r 2 : 123. π arctan l : 124. l l2 cos φ (b+d) 0 , ee φ > arccos d+2r : 125. , ee b + d < L sin α 1, ee l ( √ √L sin α v ) r v 2 b+d ≥ L sin α: 126. 1− 1− L 1 + ( u ) , ee r 1 + ( u ) ≤ L 1, √ ee r 1 + ( uv )2 ≥ L: 127. Drami hastowyown petq kazmi nra tramag i ≈ 0, 354 mas: 128. a) 32 , b) 23 : 129. 3 ln364+5 ≈ 0, 2544: 130. 17 : 131. 25 : 132. 12 : 139. Kaxyal en: 140. a) Ankax en, b) ankax en: 143. Ankax en (Ai , Aj ), i, j = {1, 2, 5, 6} zowyger (A1 , A5 , A6 ), (A2 , A5 , A6 ) e yakner: 144. 31 96 ≈ 0, 323: 145. 120 ≈ 0, 9083: 146. a) 0,94 b) 0,38: 147. a) 0,3 b) 0,6: 148. 115 ≈ (n−l)!(n−k)! 1, ee 0, 4956: 149. 0,96: 150. 1 − n!(n−l−k)! , ee k ≤ n − l ( ) 2 π−2 4 k > n − l: 153. 5! π 4π ≈ 0, 0052: 154. a) 0, 552, b) 0, 012, g) 0, 328, d) 0, 088: 155. p1 = 23 , p2 = 13 : 156. p1 = 47 , p2 = 27 , p3 = 17 : nn

n−k

(

)

a+b b b 157. p1 = a+2b , p2 = a+2b , p2 = a+b p1 : 158. 0,455: 159. 1/6: 160. 77/165, 53/165, 35/165: 161. 2/3: 162. a) 0,0792 { , b) 0,264: , ee i = 0 163. a) 0,4 b) 1/26: 164. P (ξ = i/η = 0) = 192 : 19 , ee i = 1, 9 p 165. 8−7p : 168. a) Al Bk patahowyner ankax en cankaca

l -i k -i depqowm, b) A2 C2 patahowyner ankax en, g) A4 r ≤ 0 C4 patahowyner kaxyal en: 169. r ≥ 23 , r = 13 depqerowm: 172. a) A ajin sa ori amenahavanakan parowna{ kowyown skzbnakan parownakowyan pahpanowmn : b) 125 ≈ 0, 336: 173. Aveli havanakan , or owsano girq kgtni: 174. Apacowcel, or ee A1 ∩ A2 ∩ . . . ∩ An ⊂ A apa

P (A) ≥ P (A1 ) + P (A2 ) + . . . + P (An ) − (n − 1) : (

)

177. 0,5: 178. a) Oroneli havanakanowyown PN,R = 1 − ( Rr )3 N , 4πλr 3 b) lim PN,R = e− 3 : 179. e−λt : 180. 1 − (1 − (1 − p)m )k : N →∞

N → 34 πλ R3

181. a) 1 − (1 − p)k , zowgahe miacva dimadrowyownneri qanak avelacnelow depqowm ayi howsaliowyown aowm : b) pk , hajor{ dabar miacva dimadrowyownneri qanak avelacnelow depqowm a ayi howsaliowyown nvazowm : 182. a+b : 184. 2100100! : 185. (50!)2 2n m!n! (m+n)!

: 186. a)

n ∏

k=1 n ∑

ee pk < 1: 188. 1 − e−1 :

b) 1−

(−1)k−1 k! :

k=1

pn = 1 −

(−1)k k!

, b) 1−

n ∑

n ∏

(1−pk ),

k=1

189. pn =

n ∑

n ∏

(1−pk )

k=1

pk 1−pk

(−1)k−1 k! , lim pn =

k=1

), lim pn = e−1 : (−1)k−1 ( k!

k=1

n ∑

n→∞

191. a)

n−1 ∑ (−1)k−1 k! , g) e−1 : 192. 1− (−1)m−1 Cnm (1 k=0 k=0 ( m1 ) m=1 n+2 m2 k : 193. 7 ≈ 0, 39: 194. 1 −m ) + : 196. n 2 n1 +m1 n2 +m2 : 195. 2(n+1) ( 3 a 0,4: 197. a+b : 198. Nanakowyown owni: 199. C 3 (c+d+3) Ca (c + a+b ) 3) + Ca2 b(c + 2) + Cb2 a(c + 1) + Cb3 c : 200. ≈ 0, 0811: 201. a) ≈ 0, 5739, b) ≈ 0, 7777: 202. 10 17 ≈ 0, 5882: 203. 69 ≈ 0, 3623, 69 ≈ m!

n−m ∑

190.

(1−pk ),

0, 4058,

≈ 0, 2319: 204. 165 ≈ 0, 9697: 205. Ai -n` xmbaqanak parownakowm i xotanva artadranq, i = 0, 1, . . . , 5: Amena{ (1−γ)α havanakan A5 -: 206. 20 21 ≈ 0, 9524: 207. a) (1−γ)α+γ(1−β) , b) m−2 ≈ 0, 9173: 208. 11 ≈ 0, 4545: 209. m+n−2 : 210. 136 ≈ 0, 4615: 211.

≈ 0, 5263:

212. 11/45: 213. 0,37209: 214. 4/9 : 215. (n−m)(n−m+1) : 2n(n−1)

216. P2t (s) = ( 34 )N :

s ∑

k=0

Pt (k)Pt (s−k), P2t (s) =

(2λt)s −2λt s! e

: 217. 1/2 : 218.

219. 8 owai petq owarkel a ajin rjan, p ≈ 0, 7378: 220. 36C 31 (6abc + 3b(b − 1)c + 2c(c − 1)b + 4c(c − 1)a): 221. pn = a+b+c

(p + q − 1)n−1 p1 + (1 − q)[1 + (p + q − 1) + (p + q − 1)2 + . . . + (p + q − 1−q β2 α2 1)n−2 ], lim pn = 2−p−q : 222. a) P (A) = 1−2αβ , P (B) = 1−2αβ , n→∞

orte P (A)-n P (B)-n hamapatasxan A B xaaconeri amboj xa tanelow havanakanowyownnern en: b) Qani or α > β , apa A xaacoi hamar ahavet xaal amboj xa: ( p )a+b −( p )b a 223. a+b : 224. q ( p )a −1q : 225. Oroneli havanakanowyown P = N! (N +1)N N

·2

N ∑ k=0

k k+1 ,

vanakanowyown

P =

] (1 − pi )(1 − pj )) :

N! < P < 2 (N +1) N :

N! (N +1)N ∑ i=0

ai N

[

ai −1 N −1 (1

226. Oroneli ha{

− (1 − pi )2 ) +

aj N −1 (1

i̸=j=1 2n+1 3n 2αp1 2αp1 +(1−α)(p2√ +p3 ) 0, 028 ( 34π3 )4

−

228. : 229. : 230.≈ 0, 0392: 231 ( n11 + n12 )/( n11 + n12 + n13 ): 232. : 233. : 236. : 237. 243 ≈ 0, 3292: 234. ≈ 0, 0002: 235. ≈ 27 ≈ 0, 2963: n 238. C2n−k ( 12 )2n−k : 239. ≈ 0, 2: 240. ≈ 0, 9308: 241. ≈ 0, 0655: 242. n > 10: 243. n ≥ 25: 244. ≈ 0, 0226: 245. ≈ 0, 2454: 246. ≈ 0, 0003: 247. 11/32: 248. a) 1/16, b)1/32, g) 5/16, d) 1/4, e)31/32 : 249. 0,7361; 0,7358 250. 0,393: 251. ≈ 0, 0013: 252. Aveli havanakan stanal gone mek <6> ors za i netowmov: 253. a) ≈ 0, 6651, b) ≈ 0, 6187, z) ≈ 0, 5973: 254. mo = 4, P10 (4) ≈ 0, 2508: 255. 24 ≤ n ≤ 25: 256. mo = 10: 257. ≈ 0, 0041: 258. ≈ 0, 0456: 259. Aveli havanakan 21 -: 260. 5 ≤ m ≤ 227.

≈ 0, 3171:

∑

22:

261. n = 666: 262. ≈ 0, 7698: 263. 15 ≤ m ≤ 33: 264. R- aha gowmarn , 1600 ≤ R ≤ 3733 13 : 266. pk + k(1 − p)pk−1 + k(k−1) (1−p)2 pk−2 : 267. a) ≈ 0, 6513, b) ≈ 0, 998: 268. ≈ 0, 9936: 269. ≈ 0, 197: 270. ≈ 0, 5953: 271. 0, 0935: 272. 256 ≈ 0, 3633: 1 2n n 273. 273. a) 0, 321, b) 0, 243: 274. C2n ( 2 ) : 275. pm (1 + Cm1 q + n−1 Cm+1 q 2 + . . . + Cm+n−2 q n−1 ): 276. Xaagowmar baanel a ajin erkrord xaaconeri ahelow havanakanowyownneri p1 : p2 ( 1 2 1 1 haraberowyan hamematakan` p1 = 2m 1 + 2 Cm + 22 Cm+1 + ) n−1 · · ·+ 2n−1 Cm+n−2 , p2 =

277. 281. k

(

−λp : 279. 1 − (1 − p(1 − q))n : 278. (λp) e! e B -n -navi xortakowm , Ak -n - navin kdipi k torped, n n ∑ ∑ ): = 0, n:P (B) = P (Ak ) · P (B/Ak ) = Cnk pk q n−k (1 − mk−1 2 (1

+ (q − p)n ):

) m−1 1+ 12 Cn1 + 212 Cn+1 +· · ·+ 2m−1 Cm+n−2 :

k=0

n ∑

k=2

282. 1 − (1 − p1 + , orte l = [ Mk ]: m2 =l+1 283. A-n mrcman amanak khai hraigneric miayn mek, Ai -n mrcowm khai i-rd hraig: A(m) - i-rd hraig owni m dipowm, i isk mnaca hraigneric yowraqanyowr o aveli, qan (m − 1) dipowm, Pm,n (i)-n i-rd hragi m dipowm stanalow havanaka{ nowyownn , Tm (j) -n j -rd hragi o aveli, qan (m − 1) dipowmner n ∑ (m) stanalow havanakanowyownn , P (Ai ) = P (Ai ), P (A) = )n

k ∑

P (Ai ) =

k ∑ n ∑ i=1 m=1

i=1

m2 n−m2 n! m2 !(n−m2 )! p2 p3

Pm,n (i) ·

∏ j̸=i

n+m

m=1

Tm (j):

k−1 k l−k 284. Cl−1 p q :

- amboj iv 285. : 286. Cn p 2 q 2 , ee n+m 0 haka ak depqowm: 288. a) pq3 , b) (1 − q3 )pq3 , z) (1 − q 3 − pq 3 )pq 3 : 289. q ; 12 : 290. q 2 ; 0, 25: 291. 1290 ≈ 0, 0054: 292. a−1 b p (1−q ) N r−1 : 293. (1 − p ) : 294. (rpq + qr )N : 296. ≈ pa−1 +q b−1 −pa−1 q b−1 Cnm pn+m q n−m

∑

n+m

n−m

1 k 9 1000−k k ) ( 10 ) ≈ 0, 9993: 298. a) ≈ 0, 4236, 297. C1000 ( 10 k=0 b) ≈ 0, 5, z) ≈ 0, 48: 299. n = 100: 300. 0 ≤ m ≤ 6, orte m - baxowmneri ivn : 301. a) 558, b) 541: 302. ε = 0, 05: 303. 547:

0, 265:

0 1 2 1/4 1/2 1/4 305. ξ -n standart manrakneri ivn , Pξ 1/45 16/45 28/45 306. ξ -n xotan manrakneri ivn , ξ P 0,6561 0,2916 0,0486 0,0036 0,0001 307. ξ -n stowgva sarqeri ivn , ξ P 0,1 0,09 0,081 0,0729 0,6561 308. Ω = {g, g, g, ..., ... g, ...}, ξ -n netowmneri ivn , P (ξ = | {z } 304. ξ -n gerbi handes galow ivn ,

ξ P

n−1

, n = 1, 2, ..., P (ξ > 1) = 12 : 309. ξ -n katara krakocneri ivn , P (ξ = k) = 0, 8k−1 0, 2, k = 1, 2, . . .: 310. ξk -n k-rd bas{ ketbolisti katara netowmneri ivn , k = 1, 2: P (ξ1 = m) = (0, 6 · 0, 4)m−1 (0, 4 + 0, 62 ), m = 1, 2, . . ., P (ξ2 = 0) = 0, 4, P (ξ2 = m) = 0, 6(0, 4· 0, 6)m−1 (0, 6 + 0, 42 ), m = 1, 2, . . .: 313. 0,25: 315. n) =

 0, F (x) = 34 x −   1,

x ≤ −1 3/4, + −1 < x ≤ 1 : 316. a) Hnaravor , b) x>1 o, g) hnaravor : 317. a) a = 1/π, b) F (x) = 12 + π1 arctan x, g) P (−1 ≤ ξ < 1) = 12 : 318. a) P (ξ ≥ 1) = 14 , b) P (|ξ| ≥ 1) = 12 : 319. a = π2 , π42 (arctan e)2 : 320. a) c = 2/a, b) F (x) =   x≤0 0, ( ) x x 0 < x ≤ a g) P ( a2 ≤ ξ < a) = 14 : 321. a) c = 1/a, a 2− a ,   1, x > a,   x ≤ −a  0,   1 + x + x2 , −a < x ≤ 0 b) F (x) =  21 xa 2ax22 g) P ( a2 ≤ ξ < a) = 18 : 323. + − , < x ≤ a  a 2a2   1, x > a, x3

{

e−1

, b)

η1 P 326. Pη fη (x) =

e−1

, g)

e−1

: 324. P (ξ−t < x/ξ ≥ t) =

1 − e−λx , 0,

1 2 5 η2 0 1 2 0,3 0,5 0,2 , P 0,3 0,5 0,2 : : 327. a) fη (x) = 2√1 x , F (0) 1 − F (0)

325.

, x > 1{, g) fη (x) = x1 , 1 < x < e:

x ≥ 0, x<0:

0 < x ≤ 1,

{ 1 − e−x , x ≥ 0, e−x , x≥0 328. a) Fη (x) = fη (x) = 0, x < 0, 0, x < 0, ( x ) x e b) Fη (x) = Fξ 1+ee −x , fη (x) = (1+e x )2 , { { 1 − e−λx , x≥0 λe−λx , x≥0 g) Fη (x) = fη (x) = 0, x < 0, 0, x < 0, { √ , x ∈ (−1, 1) , 329. a) fη (x) = π 1−x2 0, x∈ / (−1, 1),  {  x≤0 0, √ , x ∈ (0, 1) π 1−x b) fη (x) = 330. Fη (x) = x, 0<x≤1 0, x∈ / (0, 1),  1, x>1:   x≤0 0, 331. Fη (x) = x, 0 < x ≤ 1 332. fη (x) = π(1+x 2) :  1, x>1:  x≤0  0, √ 2 −4x l 333. F (x) = 1 − l , 0 < x ≤ l2 /4  1, x > l2 /4, { √ 2 , x ∈ (0, l2 /4) f (x) = l l2 −4x 0, x∈ / (0, l2 /4) :  x≤0  0, (a−x)2 334. F (x) = 1 − a2 , 0<x≤a  1, x > a,

{ f (x) =

2(a−x) , a2

335. Nanakenq fξk (x)-ov,

x ∈ (0, a) x∈ / (0, a) : axic k-rd keti

{

fξk (x) = {

336. a) f (x) = {

337. f (x) = {

338. f (x) =

k−1 1 Cn1 Cn−1 a 0,

1−

2a , π(a2 +x2 )

x > 0,

x ≤ 0,

√ x , 2R 4R2 −x2

0, √ 1 , π R2 −x2

( x )k−1 (

abscisi xtowyown ) x n−k a

x ∈ (0, a) x∈ / (0, a) :

b) f (x) = π(a2a+x2 ) :

x ∈ (0, 2R), x∈ / (0, 2R) : x ∈ (−R, R), x∈ / (−R, R) :

339. 1/2, 1/6, 1/12, 1/2: 340. a) 1/3, b) 1/3: 341. 0,3; 0,4; 0,5: 342. 0,0456: 343. a) 0,79673 b) 0,68268 g) 0,3707 d) 0,95154 e) 0,15866: 348. η-n m −λp , m = 0, 1, 2, . . .: hajo oreri ivn , P (η = m) = (λp) e m! { 353. Karo : 354. η = 12 (ξ + |ξ|), Fη (x) = 355.

356. 357.

 ( ) y−b  F   a( ,  )   1 − F y−b + , a Fη (y) =   0,    1,

  x≤0 0, Fη (x) = x, 0<x≤1   1, x>1:   x ≤ πa2  0, √ x/π−a Fη (x) = πa2 < x ≤ πb2 b−a ,   1, x > πb2 :

F (x), 0,

x>0 x≤0:

a>0 a<0 a = 0, y ≤ b, a = 0, y > b

359. P (signξ {= 1) = P (signξ = −1) = 1/2: √ 1 e−x/2 , 2πx

360. fη (x) = 361. a)

x>0

0, x≤0: ( √ )  √ ( x−a)2 ( x+a)2 − −  √1 2σ 2σ e +e , fη (x) = 2σ 2πx  0,

b) fη (x) = 3σ√2π1 √3 x2 e {

362. fη (x) =

1, 0,

363. c = 1, fη (x) =

√ ( 3 x−a)2 − 2σ 2

x>0 x≤0:

, x ̸= 0:

x ∈ (0, 1) x∈ / (0, 1) : { √1 , 2 1−x

x ∈ (0, 1)

0, x∈ / (0, 1) : (1 )( ) x 364. c = , F (x, y) = π{arctan 4 + 12 π1 arctan y5 + 12 : (ln2 2)2−x−y , x ≥ 0, y ≥ 0, 365. 1) 128 , 2) f (x, y) = 0, mnaca depqerowm, −12 g) 135 · 2 : 366. a = π2 , fξ (x) = fη (x) = π(1+x η -n ankax 2 ) , ξ -n π2

en: 367. a) fξ (x) = e−x , x ≥ 0, b) fη (y) = (1+y) 2 , y > 0: 371. 14/39, 10/39, 10/39/ 5/39: 372. a) pF (x) + qG(x), b) F (x)[p + qG(x)], g) F (x) + qG(x)[1 − F (x)]: 374. O: 375. Ayo: 377. O: 378. O: 379. a) Ayo, b) ayo, g) ayo, rinak` ξ -n ndownowm -1 1 areqner, isk η = −ξ : Ayd depqowm ξ + η{= 0 ξη = −1: 382. Ayo: ndhanrapes

k oxvi: 388. fξ,η (x, y) = {

1/2, 0,

(x, y) ∈ R , (x, y) ∈ /R

1 − |x|, |x| ≤ 1 fξ (x) = fη (x) = ξ -n η -n kaxyal en: 0, |x| > 1, { √ 2 2 2 R −x , |x| ≤ R, πR2 389. fξ (x) = 0, |x| > R, { √ 2 2 { 2 R −y , |z| ≤ H, , |y| ≤ R, πR fη (y) = fζ (z) = 2H 0, |z| > H, 0, |y| > R, (ξ, η, ζ) vektori proyekcianer kaxyal en:

{

390. a) fξ (x) = fη (x) = fζ (x) =

r2 −x2 , r3

393. a) F (x, y) = F (min(x, y)), b)

  0, F (x, y) = F (x) − F (−y),   F (y) − F (−y),

|x| ≤ r |x| > r,

b) 1/8:

x ≤ −y, y ≤ 0 −y < x < y, y > 0, x ≥ y, y > 0 :

394. b) (12x2 + 6x)/7 g) 15/56 d) 0,8625: 397. P (ηn = 1) = P (ηn = 3 2 x , fη (x) = −1) = 1/2: 398. 5/6, 1/6: 399. a) 1/8, b) fξ (x) = 38 − 16 y 1 −y 3 1 3y −a n 4 e y : 400. 1/2, 1−e : 403. 2 + 4x − 4x3 : 404. Fξ (x) = 1−(1−F (x)) , n ∑ Fη (x) = (F (x))n : 405. Fη (x) = 1 − exp{−( λi )x}, x ≥ 0: i=1

0,06 0,15 0,32 0,27 0,2 ξ−η -3 -2 -1 P 0,08 0,18 0,35 0,24 0,15 ξη -2 -1 P 0,08 0,06 0,51 0,15 0,2 η 407. ξ + P 0,08 0,16 0,28 0,24 0,24 ξ−η P 0,12 0,24 0,32 0,16 0,16 ξη -2 -1 0 P 0,16 0,16 0,2 0,24 0,24 408. P (η = i) = 2qi p − q2i p − q2i+1 p, P (η = i, ξ1 = j) = qi+j p2 , i > j , P (η = i, ξ1 = i) = (1 − q i+1 )q i p, q = 1 − p: { 406.

ξ+η P

(F (y))n − (F (y) − F (x))n , x<y n (F (y)) , x≥y: { k+1 , k = 0, 1, ..., n, (n+1)2 P (ξ + η = k) = 2n+1−k , k = n + 1, ..., 2n : (n+1)2

409. F (x, y) = 418.

422. fη (x) =

   0,

x−2a , (b−a)2    2b−x2 , (b−a)

{ (

423. a) fη (x) = b) fη (x) =

x ≤ 2a, x > 2b 2a < x ≤ a + b

(Simpsoni baxowm)

a + b < x ≤ 2b : ) 1 − |x| , |x| ≤ a a

0, ) { ( |x| − , a a

|x| > a, |x| ≤ a

|x| > a,   x∈ / [0, 2] 0, 424. a) fη (x) = x, 0<x≤1  2 − x, 1 < x ≤ 2, { 1 − |x|, |x| ≤ 1 b) fη (x) = 0, |x| > 1, { 2(1 − x), 0 < x ≤ 1 g) fη (x) = 0, x ≤ 0, x > 1 { − ln x, 0 < x < 1 d) fη (x) = 0, x < 0, x ≥ 1   x<0 0, e) fη (x) =  2 , 0<x≤1  1 x>1: 2, 2x  x ≤ −1, x > 3 0, x+1 425. fξ+η (x) =  4 , −1 < x ≤ 1  3−x 1<x≤3: 4 , 0,

−|x| : 426. fη (x) = 1+|x| 4 e

427.

  0, fξ+η (x) = a1 (1 − e−λx ),   1 −λx λa (e − 1), ae

x<0 0 ≤ x ≤ a, x>a:

{

430. a) fη (x) = {

b) fη (x) = 433. fη (x) =

x ≥ − λ1 ln 2, x < − λ1 ln 2

, (1+x)2

x ≥ 0,

x<0:

π(1+x2 )

{ 1, 436. fη (x) = 0,

438.

λ −λx , 2e

431. fξη (x) = 12x(1 − x)2 , x ∈ (0, 1):

: 435. Fη (x) =

x ∈ (0, 1) x∈ / (0, 1) :

  0, fη (x) = x2 ,   x(2 − x),

{

1 − x1 , 0,

437. fη (x) =

x∈ / (0, 2] 0<x≤1 1<x≤2:

x>1 x≤1:

  0,

x∈ / (0, 1] 0 < x ≤ 1/2

, 2(1−x)2   1 , 2x2

1/2 < x ≤ 1 :

439. fξη (x) = 12x(4x − x2 − R2

2 ln x − 3), x ∈ (0, 1): 440. fξη (x) = e−x , x > 0: 443. 1 − e− 2 : 444. η ∼ N (0, 1): 448. Eξ = 0, Dξ = n(n+1) : 449. ξ -n dipowmneri ivn : Eξ = 14, Dξ = 4, 2: 450. p = 2/3, n = 18: 451. $5: 452. ξ -n vercra artadranqneri mej gtnvo xotan artadranqneri ivn , Eξ = 3/5, Dξ = 28/75: 453. ξ -n hanva gndikneri mej spitak gndikneri ivn , Eξ = 0, 8, Dξ = 0, 36: 454. n(2p − 1): 456. k = a, A = e−a : 457. Eξ = a, Dξ = a(a + 1): 458. a = 1/10 , an q = 9/10: 459. a) Dξ = Eξ(Eξ + 1), b) P (ξ = n) = (1+a) n+1 , n = 0, 1, ... (Paskali baxowm): 460. ξ -n netowmneri { ivn , Eξ = 2, 1/6, x ∈ (1, 7) Dξ = 2: 461. Eξ = p1 , Dξ = 1−p : 462. f (x) = p2 0, x∈ / (1, 7) : √

463. a = λ/2, Eξ = 0, Dξ = λ22 : 464. A = 2λ2 , Eξ = 2λπ , Dξ = 4−π : 4λ2 465. Eξ = n + 1, Dξ = n + 1: 0 1 b) Eη = 2/3, Dη = 2/9: 467. Eη = 2, 4, 466. a) Pη 1/3 2/3 λ λ Dη = 1, 99: 468. Eη = 1+λ , Dη = (2+λ)(1+λ) 2 : 469. a) 1/2, b) e − 1: 470. a) Eη = 0, Dη = 1/3, b) Eη = 1/2, Dη = 1/12: 471. a)

1/2, b)

−1/2:

472.

Eη = l/4, Dη =

ln 2:

473.

  0, Fη (x) =

x≤0 0 < x ≤ l/2 , x > l/2,

2x ,  l



48 : 474. Eζ = 1, 9, Dζ = 1, 29: 475. 1) 2/17, 2) 35/12, 3) 4,87, 4) 227/33, 5) 43/83, 6) 35/33: 476. b) Eξ = 7/2, Eη = 161/36, Dξ = 35/12, Dη = 2555/1296√, cov(ξ,√η) = 105/72:√ 477. Eζ = −6, 2 −β 2 4 6 Dζ = 29: 479. αα2 +β 2 : 481. a) 4 , b) 5 : 482. a) π 2 , b) 0: 483. a) 18 b) 21: 484. a+1/k, 1/k2 , b) 1, 1/6: 485. Dξ = 1, Dη = 1, cov(ξ, η) = C k C m−k 0: 486. 2, 5/3, 12/5: 487. a) P (ξ = k) = n CNm−m , k = 0, 1, ..., m, b) N mn(N −n)(N −m) mn Eξ = N , Dξ = : 488. Eξ1 = 4, Eξ2 = 3: N 2 (N −1) 1−(2p−1)n : 490. ξ -n hanva s gndikneri ivn , Eξ = 489. Eξ = n/m: 491. P (ξ = k) = q k p + pk q , k = 1, 2, ..., p-n mek orowm hajoowyan havanakanowyownn , q = 1 − p: Eξ = pq + pq , Dξ = −λ p + pq2 − 2: 492. Eξ = α/β , Dξ = α/β 2 : 493. 1−eλ : 496. c = 2, q2 √ √ Eη = 1/2, Dη = 1/12: 497. a) σ π2 , b) σ π2 : 498. Eρ = a/3, Dρ = a2 /18: { 499. Eη = 23 R, Dη = R182 :

500. Fξ (x) =

goyowyown owni:  501. Fξ (x) =

arctan

x R,

arcsin

x 2R ,

 0, π



x≤0 fξ (x) = x > 0,

2R π(R2 +x2 )

x ≤ 0, 0 < x ≤ 2R, Eξ = x > 2R,

4R π

, x ≥ 0, Eξ -n

504. Eη = e , Dη = 12 (1 − e−1 )2 : 506. 155: 507. Eξn = 1, Dξn = 1: 510. Eη = 0: 514. (σ1 − σ2 )2 ≤ D(ξ + η) ≤ (σ1 + σ2 )2 : 516. Eξη = 1, Dξη = 4/9√ : 517. Eζ√ = 1/4, Dζ = 7/144: 518. Eζ = 0, Dζ = 1/24: 519. a = − 3, b = 3: 521. Eξ k = λk!k : 522. µ2k+1 = 0, µ2k = (2k − 1)!!σ2k ( 29 ) η \ ξ -2 -1 1 2 0 1/4 1/4 0 527. B = 0 1 : 529. a) o, 4 1/4 0 0 1/4 − 21

(

) σ 2 3σ 4 b) ayo: 530. -1/2: 531. B = 3σ4 15σ6 : 533. n−m n : 534. ρ : 538. y2 /3: 539. λ1nλ+λ1 2 : 540. 5−4y 8−6y : 541. y + 2: 542. 2: 543. 1/8: 544. n 1, 1, 1: 545. -23/6: 546. 6, 112/33: 547. n+1 , n+1 : 548. Eξ = 1,

Dξ = 1:

−m−1) 549. (2N −m)(2N : 552. 14 : 553. 89 : 554. 0,64: 555. 2(2N −1) 0,909: 557. > 23 : 558. > 0, 84: 559. ≤ 11 31 : 560. ≥ 19/20: 561. √ p1 = 0, 75: 562. > 4 ; > 40 : 563. 0,94: 564. 0,75: 565. ε = 302 : 566. ε = 0, 3: 567. a) ≤ 15 17 b) ≥ 4 g) n ≥ 10: 568. ≤ 48 ; 0,2: 571. Ayo: 572. Ayo: 573. Ayo: 574. Ayo: 575. Ayo: 579. n ≥ 250000: 580. p > 0, 82: 581. p > 0, 98: 582. S ≥ 815: 584. p > 0, 9875: n ∏ 585. s < 1/2: 591. cos t: 592. cos2 t: 593. (qk +pk eit ); qk +pk = 1: k=1

2 at

594. peita + (1 − p)eitb : 595. sinatat : 596. 4 sin : 597. a2a+t2 2 : 598. a2 t2 eitb −eiat bk+1 −ak+1 −3 1−it ; νk = k!: 599. it(b−a) , (k+1)(b−a) : 600. e : 601. a) P (ξ = −1) = P (ξ = 1) = 21 : b) P (ξ = −2) = P (ξ = 2) = 14 , P (ξ = 0) = 12 : ( 2 b a )2 (eit 2 −eit 2 ) : 603. e−|t| : 604. (q +peit )n ; np; npq : 605. 602. i(b−a)t ; a; a(1 + a): 606. µ2k+1 = 0; µ2k = σ 2k (2k − 1)!!: 607. (1 − 1+a(1−eit ) it −λ λ(λ+1)···(λ+k−1) : 609. f (x) = π(a2a+x2 ) : 610. f (x) = 21 e−|x| : ; a) ak 618. Binomakan baxowm (n+m, p) parametrov: 619. Powasonyan baxowm λ1 + λ2 parametrov: 625. a) Normal baxowm (0, 2) pa{ {

1/2, x ∈ [−1, 1] 0, x ∈ / [−1, 1] : { ( ) 0, x < 0 627. f (x) = 2π1 1+(x−1) 628. f (x) = −x 2 + 1+(x−1)2 : e , x≥0: 630. n ≥ 4: 633. 0,1699: 634. 0,04: 635. n = 450000: 636. 0,9759:

rametrov: 626. f (x) =

637. Ayo: 638. O: 644. O:

BOVANDAKOWYOWN

Xmbagri komic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Gor oowyownner patahowyneri het . . . . . . . . . . . . . . . . . . . . 4 Hamakcowyown (Kombinatorika) . . . . . . . . . . . . . . . . . . . . . . . . . 9 Havanakanowyan dasakan sahmanowm . . . . . . . . . . . . . . . . 13 Erkraa akan havanakanowyownner . . . . . . . . . . . . . . . . . . . 24 Paymanakan havanakanowyown Patahowyneri oreri ankaxowyown . . . . . . . . . . . . . . 28 Lriv havanakanowyan Bayesi bana er . . . . . . . . . . . . 39 Be nowlii bana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Patahakan me owyown baxman fownkcia . . . . . . . . . . . 61 Bazmaa patahakan me owyown baxman fownkcia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Patahakan me owyownneri ankaxowyown . . . . . . . . . . . . . . 77 Paymanakan baxowmner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Patahakan me owyownneric fownkciayi baxowm . . . . . . 85 Patahakan me owyan vayin bnowagriner . . . . . . . . 95 Patahakan vektori vayin bnowagriner . . . . . . . . . . 97 Paymanakan maematikakan spasowm . . . . . . . . . . . . . . . . . 98 Me veri renq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Bnowagri fownkcia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Kentronakan sahmanayin eorem . . . . . . . . . . . . . . . . . . . . . . 126 Ayowsak 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Ayowsak 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Patasxanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆ

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Տպագրված է «Գևորգ-Հրայր» ՍՊԸ-ում: ք. Երևան, Գրիգոր Լուսավորչի 6

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ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

Ն. Գ. ԱՀԱՐՈՆՅԱՆ | Ե. Ռ. ԻՍՐԱԵԼՅԱՆ

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ

ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆ

Ն. Գ. ԱՀԱՐՈՆՅԱՆ

Ե. Ռ. ԻՍՐԱԵԼՅԱՆ

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ

ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

ԵՊՀ ՀՐԱՏԱՐԱԿՉՈՒԹՅՈՒՆ

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Ս. Մ. ՆԱՐԻՄԱՆՅԱՆ

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XMBAGRI KOMIC

PATASXANNER

BOVANDAKOWYOWN

ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆ

ՆԱՐԻՆԵ ԳԵՎՈՐԳԻ ԱՀԱՐՈՆՅԱՆ,

ԵՊՐԱՔՍՅԱ ՌՈՒԲԻԿԻ ԻՍՐԱԵԼՅԱՆ

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ

ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ ՏԵՍՈՒԹՅԱՆ ԽՆԴՐԱԳԻՐՔ

Ն. Գ. ԱՀԱՐՈՆՅԱՆ | Ե. Ռ. ԻՍՐԱԵԼՅԱՆ

ՀԱՎԱՆԱԿԱՆՈՒԹՅՈՒՆՆԵՐԻ

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