А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 8
Текст из файла (страница 8)
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H P CR MP#`U^!¯"=8:DC¾xSa xCE @ T ]j\*Q P (Z) = 1nUem*N Z NP]^JlSPRTOHILMNP]^JWIEHNj\*_:VzkHQH]PNj\nJIz]j\=bk=D&dHO=DKuWNj\*QHkHQHO=DzO=Dst_-W.DN^J]KB*><'\!*=1 ¾xSa xCE £HCED1ZEQHY]PQHGHIW.D&OHOHIRËWNPGHIKJOHI]^JOHIRýFHGHI]^JG=D&OH]^J.WN]j\=bk=D&dHO=DK'WNj\*QHkHQHO=DItm*OHI&stO=DLkHOHI5IFHGHNjm*Nj\=KHN^J5]PWING=D&]PFHGHNjm*Nj\*NtOHQHN&n=OHIFHISsLDoPξ (B) = P(ξ ∈ B).(7.2)ξξξmD&OHOHIR¦bSG=D&]PFHGHNjm*Nj\*NPO=QHfOHNj\*l&sPKItm*OHI&stO=DkHOHIWI]P]^JD&OHIWQJlE]j\=bk=D&dHObfWNj\*QokHQHObT]j\*Q'FHNPGHWIN%bJWNPGM5m*N^OHQHNsLD&RTNPk=D&OHQK5m*I]^J&DoJIHkOHIzIkHNPWQm*OHI5WWQm=b"Itm*OHI&stO=DLkHOHIduIFHGHNjm*Nj\*NPOo1η(ω)OHI]^JQ;FHG=D&W&Id»k=D&]jJQû jnJI¯W&JIGHINgJGHNPv&bN^JOHNPYI&JIGHIUPI FHILKH]POHNPOHQK¸Á¦D&]P]PRTI&JGHQHR FHGHQHRTNPGζ(ω)b]^Jl sLDmD&OHI WNPGHILKJOHI]^JOHIN FHGHI]^JG=D&OH]^JWInUrm*N {;RTNPG=DEÃXNPvNPU^DÁD&]P]PRTI&Joω0 1 1([0, 1], B[0, 1] , λ)1GHQHRÀm*WNqZbOHYHHQHQ λ Qn-IFHGHNjm*Nj\*NPOHOH_-N4 2η(ω)ζ(ω)JD&Yn*YD&YuFHIYDsLD&OHI5O=D5GHQH]^bOHYN ÄSIYDM5NPR-n&kJIXG=D&]PFHGHNjm*Nj\*NPOHQK9]j\=bk=D&dHOH_:V9WNj\*QHkHQHO Q ]PIWF=DmD&f[J&ÄE\=Kη ζ|^JIUPI+G=DstIvlNPR oÓD\*UPNPvGbO=DukHN^J_-GHN'Y\D&]P]tDb]^Jl+Y\D&]P]WY\*fck=D&NjJσBB1W]PN9vIGHNj\*NPW]PYHQHN"RTOHILMNP]^JW.DnWY\*fck=D&f[iQHN9JIkHYHQQ nY\D&]P]WY\*fck=DoN^JuW]PNvIGHNj\*NPW]PYHQHNzRTOHILMNP]^JW.DnOHN5WY\*fck=D&f[iQHN 0Q 1nY\D&]P] B2WY\*fck=D&NjJW]PNXvIGHNj\*NPW]PYHQHNRTOHILM5NP]jJW.DnWY\*fck=D&f[iQHN QOHNX0WY\*1fck=D&f[iQHNB3 n*Y\D&]P]WY\*fck=D&N^JW]PN[vIGHNj\*NPW]PYHQHNRTOHIMNP]^JW.DnOHNcW0Y\*f[k=D&f[iQHN Q9WY\*fc1k=D&f[iQHN B4¤kHNPWQm*OHIHnkJIQntÄE\=K0\*fcvIUPI[RTOHILMNP]^JW.1DB = B1 ∪B2 ∪B3 ∪B4 Bi Bj = ∅ i 6= j]PFHG=DWNjmH\*QHWI9G=D&WNPOH]^JWI[Ïd»¼±ºVä¼¾ ¤[B1 ∈ B 1P{ω| ζ(ω) ∈ B1 } = λ([0, 1]\{1/4}) = λ([0, 1]) = P(η ∈ B1 ) = 1.xcO=D\*IUPQHkHOHIHnmH\=KnBi ∈ Bi i = 2, 3, 4n=QHRTNPNPRP(ζ ∈ B2 ) = λ({1/4}) = λ(∅) = P(η ∈ B2 ) = 0b!cWde(fhgi ã¢Q1P(ζ ∈ B3 ) = P(ζ ∈ B4 ) = P(η ∈ B3 ) = P(η ∈ B4 ) = .2 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jnIFHGHNjm*Nj\*NPOHOHIdO=D[WNPGHILKJOHI]^JOHIRqFHGHI]^JG=D&OH]^JWN([0, 1], F,n=ξUem*NnQP)F = {Ω, ∅, [0, 1/2], (1/2, 1]} P([0, 1/2]) = 0.1 P((1/2, 1]) = 0.9¾x%ÄSxTx» Ö *IYDsLDJlHn*kJI"NP]j\*Q+]j\=bk=D&dHO=DK+W&Nj\*QHkHQHO=D QHRTNPN^J"ZbOHYoξHQHfÂG=D&]PFHGHNjm*Nj\*NPOHQKnTQstIv&G=DM5NPOHObfÂO=DGHQH]^bOHYNu @ nJImH\=KP(ξ = x) = 0W]PNjVx ∈ IR¾x%ÄSxTx» L*IYDsLDJlHn*kJI"NP]j\*Q+]j\=bk=D&dHO=DK+W&Nj\*QHkHQHO=D QHRTNPN^J"ZbOHYoξHQHfG=D&]PFHGHNjm*Nj\*NPO=QKn.QstIvG=DMNPOHObfO=DGHQH]^bOHYNÔ¢nJImH\=KFHIokJQ'W]PNjVqFHIRTNPGHNcÃXNPvNPUjD. nOHI]^biNP]^J.W&b.N^J5JD&YIN P(ξnk=JI x) = 0x ∈ IRxP(ξ = x) 6= 0¾x%ÄSxTx» & *IYDsLDJlHn*kJIzmH\=Ky\*fcvId]j\=bk=D&dHOHId+WNj\*QHkHQHOH_QHRTNPN^JξRTNP]^JI9G=D&WNPOH]^JWImH\=KuFHIkJQW]PNjVFHIzRTNPGHNÃXNPvNPUjD. ["HEA!><"HE6[P(ξ = x) = 0x ∈ IRsã t × w Æß.ÞàHv²c«ª« w ®ß&®Hv²1®u® Î(x Þ.Ù Øx «¬« w «H®vyø ùûúýüËþ¿ÿ æ b]^JlzsLDmD&OHIzOHNPYI&JIGHINXWNPGHILKJOHI]^JOHINSFHGHI]^JG=DOH]^J.WI(Ω, F, P)¤EÁ Ä ¦Ã aCE Ô¢- B'HE? - +H ç / |q-DCnHE><>U\1! è!*^O>U#B'!!><}'!! 7 ξ9!8:;+-$+ B^M)"'==-DC"><*1U CP (B) = 1'!><'!DC5"R5=8:A!)+;+B$+!'DC7^M-)*'=-"!$&+ VHE6R ?-,)'=-"*1nBHE><1!*1n.¾xSa xCE Ô¢OHIUemDUPIWIG.KJn&kJI]j\=bk=D&dHO=DK5WNj\*QHkHQHO=DQHRTNPN^Jm*QH]joYHGHN^JOHINXG=D&]PFHGHNjm*Nj\*NtOHQHN&nNP]j\*QIO=DFHGHQHOHQHRaD&N^J"]EOHNPObL\*NPW_-RTQ+WNPGHILKJOHI]^JLKHRTQξHO NzvI\*NPN&nkHNPR1]PkHN^JOHIN9kHQH]j\*IustO=DLkHNPOHQHd¤[m*O=D&YIHnsLD&RTN^JQHR-nkJIyJ&D&YINzIFHGHNjom*Nj\*NPOHQHNOHNqKHW\=KHN^J]K¯WFHI\*OHN YIGHGHNPYJOH_-R-ncFHI]PYI\*lYbNPR¦bbLm*IW\*NjJWIGKHN^JnJIZbOHYHHQKG=D&]jo\*fcv.DKq]j\=bk=D&dHO=DKWNj\*QHkHQHO=DnJD&YYD&Y NP]j\*QP(ξ = x) > 0FHGHNjm*Nj\*NPOHQKQHRTNPN^J¨WJIkHYNG=DstGH_-WH%CI§ZbOHYHHQKêG=D&]PFHGHNjm*Nj\*NPOHQK¯OHNxRTILMN^JzQHRTN^Jl9Fvξ I\*NPN&n=kHNPR¨]PkHN^JOHINXkHQH]j\*I5JIkHNPYG=DstGH_-W.D]PR-HsLDmDLkbu £ jÁD&]PFHGHNjm*Nj\*NtOHQHN"m*QH]PYHGHN^JOHIdg]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ÀbLm*IvOHI)FHGHNjm*]^JD&W\=KJl]EFHIRTI&ilf1J&D&YuO=Dst_-W.D&NPRTIUPIqh"U^!)VA)+B!'<^x1p1x2p2x3p3......éZ§¨¨Á˧伾 £xnpn......¾*m*NP]PlSkHQH]j\D{;|^JIWIstRTIM9OH_-N[k=D&]^JOH_-NcstO=DLkHNPOHQK"]j\=bk=D&dHOHId"WNj\*QHkHQHOH_ξnmH\=K xi jn*DSmH\=KukHQH]PNj\]PFHG=D&WNjmH\*QHW_1G=D&WNPOH]^JW.Dxi 6= xji 6= jSpipi = P(ξ = xi ) > 0,Xpi = 1.i¾ x axCE .mD\*lOHNPdpNPR`FHGHQ5G=D&vI&JN%]m*QH]PYHGHN^JOH_%RTQ9G=D&]PFHGHNjm*Njo\*NPOHQKHRTQuO=D&RgsLDLk=D&]^J&bf¥v&btm*N^JzbLm*IvO=D5QHOJNPGHFHGHN^J&D&HQK+WNPGHILKJOHI]^JQYD&YyRaD&]jo]P_Á¦Km5G=D&]PFHGHNjm*Nj\*NPO=QKnZXD&YJQHkHNP]PYHQnIFHGHNjm*Nj\=KHN^JE]PQH]^JNPR¦bJIkHNPYO=DcFHGKHRTIdnQHRTNPf[iQVuYIIGm*QHO=DJ_n=W5YIJIGH_:V]PI]PGHNjm*I&JI&kHNtOH_RaD&]P]P_xipi¾x%ÄSxaxä IYDsLDJlHnkJIG.Km`G=D&]PFHGHNjm*Nj\*NPOHQHKW&sLD&QHRTOHIuItm*OHI&stO=DkHOHIIFHGHNjm*Nj\=KHN^Jzm*QH]^YHGHN^J.OHINXG=D&]PFHGHNjm*Nj\*NtOHQHN&ÁaCE -*b]^JlsLDmD&O'OHNPYI&JIGH_-duG.KmuG=D&]PFHGHNjm*Nj\*NPOHQK*CNIUPG=D&OHQHkHQoW.DK¨Iv&iOHI]^JQG=D&]P]^bM5m*NPOHQHdn:FHGHNjm*FHI\*IM9QHR-nkJI`]j\=bk=D&dHO=DKWNj\*QHkHQHO=D`RTIoMN^JzFHGHQHOHQHRaDJl9JI\*lYIm*W.DstO=DLkHNPOHQKh QIv_-kHOHI5J&D&YuUPIWIG.KJ9W5]j\=bk=D&N&n0 1YIUemDnQ~WIIv&iN)UPIWIoP(ξ = 0) > 0 P(ξ = 1) > 0 P(ξ ∈ {0, 1}) = 1G.Kn=Iv\D&]^JlfstO=DLkHNPOHQHd]j\=bk=D&dHOHIduWNj\*QHkHQHOH_¯KHW\=KHN^J]rKW]rK"m*NPdH]^JWQJ.Nj\*lO=DKFHG.KHRaDK* jÁDstIvlNPRvIGHNj\*NPW]PYbf oÓD\*UPNPvGbO=D"kHN^J_-GHN5Y\D&]P]tDn nom [σBB 1 B2 B3b!cWde(fhgiKêQnTYD&Yg|^JI`v_:\*I ]rm*Nj\D&OHIW IvI]POHIW.D&OHQHQ§sLD&RTNPk=D&OHQK £H¾D&RTN^JQHR-nkJIB4Hm \=K\*fcvIUPI'vIGHNj\*NPW]PYIUPIyRTOHIMNP]^JW.DWNPGHILKJOHI]^JlItm*OHIoBi ∈ B iP(ξ ∈ Bi )stO=DkHOHI IFHGHNjm*Nj\=KHN^J.]rKFHI FHG=D&WQ\=bhnnP(ξ ∈ B1 ) = P(Ω) P(ξ ∈ B2 ) = P(∅)n¦D&YHQHR¿IvG=DstIR-n¦G.Km¸G=D&]joP(ξ ∈ B3 ) = P(ξ = 0) P(ξ ∈ B4 ) = P(ξ = 1).FHGHNjm*Nj\*NPOHQHK5Itm*OHI&stO=DkHOHIIFHGHNjm*Nj\=KHN^JXG=D&]PFHGHNjm*Nj\*NPOHQHN¤vG=DJOHINbJWNPG.M5m*NPOHQHNIkHNPWQm*OHIHn.FHI]PYI\*lYbnDItm*OHI&JIkHNPkHOH_-NcRTOHILMNP]^JW.Dn.YD&Y5bMNpi = P(ξ ∈ {xi })UPIWIGHQ\*I]PlzW_pN&n=KHW\=KHf[J]rKuvIGHNj\*NPW]PYHQHRTQ¾x%ÄSxTxä cb.]^JlysLDmD&OqG.Km`G=D&]PFHGHNm*Nj\*NtOHQK]PR-JD&v\*QHb jIo]^JGHIQJl"]PIIJWN^J]^J.W&bf[iSbf¿ZbOHYHHQHf¿G=D&]PFHGHNjm*Nj\*NPOHQK[m[om ÁaCE -CN-IUPG=D&OHQHkHQHW.DK5Iv&iOHI]^JQ9G=D&]jo]^bM5m*NPOHQHdnFHGHNjm*FHI\*ILM9QHR-nkJI'kHQH]j\DIvG=Ds^b.oxif[JEWI&stG=D&]^J&D&f[iSbfFHI]j\*Njm*IW.DJNj\*lOHI]^JlH¾D&ZEQHYo]PQHGb.NPR¸OHNPYI&JIGHINXkHQH]j\*I ¤kHNPWQm*OHIHn=kJIF (x)1p3xp2p1x1 x2 0Ïd»¼±ºVä¼¾ ¦x3xF (x) = P(ξ < x) =XP(ξ = xi ) =Xpi .i: xi <xi: xi <x D&YHQHRäIv&G=DstIR-nUPG=D&ZEQHY¸m*QH]PYHGHN^JO=IdêZbOHYHHQHQ¦G=D&]PFHGHNjm*Nj\*NPO=QKQHRTNPN^J9WQmn=FHIYDsLD&OHOH_-dO=DGHQH]^bOHYN ¾D&RTN^JQHR-nkJI9m*QH]PYHGHN^JO*DKqZbOHYHHQK)G=D&]PFHGHNjm*Nj\*NPOHQHK)KHW\=KHN^J]rKq]^J&bFHNPOHk=DoJIdnJINP]^Jl+QHRTNPN^J+G=DstGH_-WWuYDM5m*IdqJIkHYN nmH\=KYI&JIGHIdnxP(ξ = x) > 0FHGHQHkHNPRnHDWI]^J&D\*lOH_:V"JIkHYDV"KHW\=KHN^J]K'Yb.]PIkoOHI'FHI]^JILFKH(xOHOHI+d0)¤c−J]PFf(x)mDn=W"P(ξk=D&]^J=OHIx)]^JQn]j\*Njm=bN^JnkJI'RTNPG=D9ÃXNPvNPUjDλ{F 0 (x) 6=0} = λ{x| P(ξ = x) > 0} = 0¾x axCE Ô&ÄE\=K5JIU^IHnkJIv_¸IFHGHNjm*Nj\*QJlG=D&]PFHGHNjm*Nj\*NPOHQ=Nc]j\=bk=D&doOHId+WNj\*QHkHQHOH_1m*I]^JDJIkHOHI"sLDmDJl"OHNPYI&JIGbf¿VD&G=D&YJNPGHQH]PJQ=YbnYI&JIG=DK+ItmHoOHI&stO=DkHOHIuIFHGHNjm*Nj\=KHN^J+G=D&]P]PRaDJGHQHW.D&NPRTINzG=D&]PFHGHNjm*Nj\*NPOHQ=N&Ç]j\=bk=D&Nm*QH]PYHGHN^JoOH_:VG=D&]PFHGHNjm*Nj\*NPO=QHdJD&YHQHRTQSVD&G=D&YJNPGHQH]PJQHYD&RTQ5RTIU^bJv_JlG.Km5G=D&]PFHGHNjm*Nj\*NjoOHQKQSZbOHYHHQKG=D&]PFHGHNjm*N\*NtOHQKI&sPMNRT_`FHI&stO=D&YIRTQHRT]KS]JD&YHQHRTQXVD&G=D&YJNjoGHQH]^JQHYD&RTQnYD&Y5F\*I&JOHI]^JlHnFHGHIQstWILm=KiDK5QVD&G=D&YJNPGHQH]^J.QHkHNP]tYDK9ZbOHYHHQHQÁD&]P]PRTI&JGHQHRO=D&QHvI\*NPNk=D]^JI'W]^JGHNPk=D&f[iQHNt]rKqWQm*_¯m*QH]PYHGHN^JOH_V)]j\=bk=D&doOH_:V WNj\*QHkHQHOQ G=D&]PFHGHNjm*Nj\*NPOHQ=d¦çÇvI\*l&pQHOH]^JWIR1QszOHQV RT_¿bM5N5W]jJGHNPk=Do\*QH]PlzWzFHGHNjm*_m=biQVF=D&G=D&UPG=D&ZXDV¤EÁ Ä ¦Ã aC q Sq|EH<>!1!*^\v'<>!'!!+;$+^+¥$9F8+)!*1ξÒ!;+$*+><= a^QÓ Cl+-$¥B'98FP(ξ)+!=!$&a) V=R 1/ R-|0;+!*>U!'9ξ +ár=HEâ ãr6â a?R -A+)+\!'"=8:¤EÁĦÃaCÔ|q!H>1!!*^LB'!><'!! '< -Ñ=-#'!>U-='E'+D BMR )'=-"A)+\#!'DC-\' P(ξξ = x ) = 1/nC x ∈ IRCii = 1, .
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