А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 12
Текст из файла (страница 12)
. . , 24WNj\*QHkHQHObn*} ]PII&JWN^J]^J.Wbf[iSbf ξonR¦b"QH]PFH_J&DOHQ=f¥ :NPGHObL\H\*Q¾D&RTN^JQHR-n=kJIom nξ=24Xξn .n=1CED&dm*NPR¸RaDJNPRaDJQHkHNP]PYINSILM9QmD&OHQHN IzZEIGHR¦bt\*N' Ö Ö jn=FHI]^YI\*lYby]PIv_:oJQKQIvG=DsPbf[J9FHI\*ObfÇUPGbFHFbξn*nQHRTNPNPRAB| A) + P(B)E(ξ | B).G=TD&]j]P\*FHQSGHFHNjm*GHINj\*QNPstOHI&pXQHN)\*Ic]]PF=ID&vG=Eξ_D&JRTQHN^=N JAG=P(A)E(ξntJIc]j\=bk=DdHO=DKSWNj\*QHkHQHO=DQHRTNPN^JvQHOHIRTQ=D\*lOHIND&RTQQnDNP]j\*Q¸FHGHIQξstI&pX\*I`]PIv_JQHN nTJInnnnBWNPGHILKJOHI]^Jl5ºb]PFHNjVDEG=D&WOKHN^J]rK 1 0.4ça\*Njm*IW.DJNj\*lOHIHn0.8122Eξn = (0 · 0.6 + 1 · 0.4) + (0 · 0.2 + 1 · 0.8) = .333ÄE\=K O=DV.IM5m*NPOHQKWI]PFHI\*l&sPbNPRT]K]PWIdH]^JWIRRaDJNPRaDJQHkHNP]PYIUPI+ILM9QmDoOHQK¤cJWN^JHh]^J&btEξm*NPOHJIWH16[M5~_7 77¾x%ÄSxax Ö £HI\*l&sPb&KH]PlIFHGHNjm*Nj\*NPOHQHNPRqRaDJNPRaDJQHkHNt]PYIUPIIM9QmD&OHQKnLFHIoYDsLDJlz]PFHG=D&WNjmH\*QHWI]^JlzsLD&RTNPk=D&OHQHd Ö £5Q Ö Ô¢¾x%ÄSxax Ö Ô¢ÄSIYDsLDJl5]PWIdH]^JW.Dy^o RaDJNPRaDJQHkHNP]tYIUPI9IM9QmD&OHQK¾x%ÄSxax Ö :b]^Jl{ OHNPI&JGHQH=DJNj\*lO=DKHNj\*IkHQH]j\*NPOHO=DK]j\=bk=D&dHO=DKWNj\*QHkHQHO=D5]EYIOHNPkHOH_-R¸RaDJξNPRaDJQHkHNP]tYHQHR¸IM9QmD&OHQHNPR-ÄSIYDsLDJlHn=kJIEξ = P(ξ ≥ 1) + P(ξ ≥ 2) + .
. . + P(ξ ≥ n) + . . . .b!cWdef0g7iFjüÖ ¢¾x%ÄSxax Ö ÔLCED&dJQERaDJNPRaDJQHkHNP]tYINILM9QomD&OHQHNc]j\=bk=D&dHOHId'WNj\*QHkHQHOH_nHQHRTNPf[iNPd'ZbOHYoHQHfG=D&]PFHGHNjm*Nj\*NPOHQHKnLQstIvG=DMNPOHObf§O=DGHQH]^bOoYN Ö @ 12Q{ ILm*QHO=DYIWI¾x%ÄSxax Ö Ö ab.]^JlG=D&]PFHGHNjm*Nj\*NPOHOH_%N]j\=bk=D&dHOHξ_-NWηNj\*QHkHQHOH_:NPGox101OHI%\*Qn&kJInn2Eξ = Eη Eξ/η = Eη/ξ Eξ/(ξ +η) =nwEη/(ξ + η) E1/ξ = E1/η¾x%ÄSxTx Ö b.]^Jl Q{¼m*QH]PYHGHN^JOH_%N9]j\=bk=D&dHOH_-N9WNj\*QHkHQHOH_ÄSIYDoξ ηsLDJlHn*kJIzNP]j\*Qu]^biNP]^J.W&bfcJQnHJIz]^biNP]^J.W&bNPJEξ EηE max{ξ, η}¾x%ÄSxTx Ö tb]^JlQLCED&dJQQEξ = 0 E|ξ| = 1E max{0, ξ} E min{0, ξ}¾x%ÄSxTx Ö aCED&FHQH]tD&OH_FHQH]PNPR-naFHGHNjm*O=DstO=DLkHNPOHOH_V¨G=DstOH_-R;Dm*GHNP]tDonJ&D&R-RTNPN^J]rKYIOHWNPGJIW]z]PII&JWN^J]^J.W&bf[iQHRTQ§Dm*GHNP]tD&RTQQH]PlRaDyW]j\=b.onk=D&dHOHIRFHIG.Km*YN9W\*IMNPOH_»WYIOHW&NPGJ_Tb]^Jl{ÑkHQH]j\*I+FHQH]PNPR-nYI&JIGH_-NFHI]j\D&OH_¯JNPR¸Dm*GHNP]tDJ&DR%n=YI&JIGH_-R¨IOHQuFHGHNjm*O=DstO=ξnDLkHNPOH_CED&dJQEξn¾x%ÄSxTx Ö Tb.]^Jl{]j\=bk=D&dHO=DK¨W&Nj\*QHkHQHO=DnTJINP]^Jl9JD&YDKu]j\=bk=D&dHO=DK+WNj\*QHξkHQHO=Dn*kJIb]^Jl{ÂNPNXZbOHYoP(ξ ≥ 0) = 1F(x)RHQKzG=D&]PFHGHNjm*Nj\*NPOHQKnQ]^biNP]^J.W&bNPJ.ÄSIYDsLDJlHnkJIEξEξ = 0+∞ (1 − F (x)) dx¾x%ÄSxTx Ö @ &b.]^Jl {¿]j\=bk=D&dHO=DK5WNj\*QHkHQHO=D]-]PQHRTRTN^JGHQHkHOH_%R¸]PR-&sLDoRTNPk=D&OHQHN" . -I&JOHI]PQJNj\lξOHI'ObL\=K+G=D&]PFHGHNjm*Nj\*NPOHQ=NPR-ÄSIYDsLDJlHnkJI9mH\=Ku\*f%ovIUPI5WN^iNP]^J.WNPOHOHIUPI QHRTNPN^J9RTNP]^JI9OHNPG=D&WNPOH]^JWIaE|ξ + a| ≥ E|ξ|¾x%ÄSxTx Ö t£Hcb.]^Jl {æ]j\=bk=D&dHO=DK WNj\*QHkHQHO=D']5YIOHNPkHOH_-R¯RaDJNPRaDJQokHNP]PYHQHR§ILM9QmD&OHQHNPR-ÄSIYDsLξDJlHnkJIEmH\=Kz\*fcv&IUPI W_-FHI\*OKHN^J]rK'OHNPG=D&WNPOH]^JWIxmax{x, Eξ} ≤ E max{x, ξ}¾x%ÄSxTx Ö L¢b.]^Jl QHRTNPN^JF\*I&JOHI]^JlHn&sLDmD&OHObf¨W[sLDmDkHN-Ô¢&CED&dJQξQEξ E(1/ξ)¾x%ÄSxTx Ö LCED&dJQ)RaDJNPRaDJQHkHNt]PYIN5IM9QmD&OHQHN]j\=bk=D&dHOHIdqWNj\*QHkHQHOH_nHG=D&]PFHGHNjm*Nj\*NPO=QHNSYI&JIGHIduIFHGHNjm*Nj\*NPOHI9WsLDmDLkHN @ ¢ξ¾x%ÄSxTx Ö CED&dJQ9RaDJNPRaDJQHkHNP]tYHQHNILM9QmD&OHQKz]j\=bk=D&dHOH_:V9WNj\*QHkHQHOηQ n=IFHGHNjm*Nj\*NPOHOH_:VWsLDmDLkHN ¢ζ¾x%ÄSxTx Ö Ö Hça\=bk=D&dHO=DK'WNj\*QHkHQHO=D G=D&WOHIRTNPGHOHIG=D&]PFHGHNjm*Nj\*NPO*DO=DSI&JoGHN^stYN~~CED&dJQ ξ nQ W\=KHf[J]rKu\*Q η QEξ Eη1 Eη21 = cos ξ η2 = sin ξ1OHN^sLD&[0,WQH2π]]PQHRT_-ηRTQ*wη2¾x%ÄSxTx Ö L*b]^Jl*IYDsLDJlHn=kJIOHNE]^biNP]^J.W&bNPJξ ∼ K(0, 1)Eξ¾x%ÄSxTx Ö & GHQm=bRaDJlSFHGHQHRTNPGzm*QH]PYHGHN^JO=IUPIG=D&]PFHGHNjm*Nj\*NPOHQK']j\=bk=D&doOHIdyWN\*QHkHQHOH_JD&YIUPIHnHkJI5]^bRTRaDnH]jJIKiDKyWFHG=D&W&Idyk=D]^JQ` Ö £ aYIOHNPkHO=DnOHIOHNE]^biNP]^ξJ.W&bNPJF (x)1Ïd»¼±ºWë¾ [à-*'!!BIE!*^[[Eξs ê<t!ý Ú ¤:«¥z9²¿Ù w Æß.ÞàHv² Î ª« w ®ß&®Hø ùûúýüËþ¿ÿ VÖ b]^Jl"sLDmD&OHI"OHNPYI&JIGHINWNPGHILKJOHI]^JOHINFHGHI]^JG=D&OH]^JWIQ+]j\=b.o(Ω, F, P)k=D&dHO=DKyWNj\*QHkHQHO=D O=D5OHNPR-ξ¤EÁ Ä ¦Ã aCEÖ Ö {,'AB'K1BEH<>!1*]1B!'<>!'!$!+;$+^ξ><'BSSDξ = E(ξ − Eξ)2 .¾ x axCEÖ HÄSQH]PFHNPGH]PQK)NP]^Jl"]PGHNjm*OHNPNYHW.Dm*G=DJQHkHNP]PYINSbY\*IOHNPOHQHNstO=DLkHNPOHQHd]j\=bk=D&dHOHId¸WNj\*QHkHQHOH_¼I&J`NPNRaDJN^RDJQHkHNt]PYIUtIILM9QmD&OHQKaD&YHQHRIvG=DstIR-nm*QH]PFHNPGH]PQK NP]^JlyRTNPG=D'G=DstvGHI&]tDG=D&]PFHGHNjm*Nj\*NPO=QK]j\=bk=D&dHOHId WNj\*QokHQHOH_¾x axCEÖ ÄSQH]PFHNPGH]PQHfÇJ&D&YMNERTILM9OHIIFHGHNjm*Nj\*QJ.l9YD&YminE(ξ − a)2 ,aFHGHQHkHNPRmD&OHOH_-d RTQHOHQHR¦bRm*I]^JQHUjD&N^J.]rKFHGHQ¦D&YHQHRIvG=DstIR-nkHQH]j\*Ia = EξNP]^Jl9O=DQ\=bkpDKIHNPOHYD5Wz]PGHNjm*OHNPYHW.DLm*G=DJ.QHkHOHIR]j\=bk=D&dHOHIduWNj\*QHkHQHOH_ Eξξ√¤EÁ Ä ¦Ã aCEÖ !@\'!><'!!!'\ VRDξ!;+$+^¾"!)#"!$& HE=*.Ä QH]PFHNPGH]PQHQS]j\=bk=D&dHOH_:VEWNj\*QHkHQHOXIv\DmD&f[J%]j\*Njm=bf[iQ=RTQXI]POHIWOH_-RTQX]PWIdoS]^JW.D&RTQh [ @ Dξ = Eξ 2 − (Eξ)2Dξ ≥ 0nHFHGHQHkHNPRDξ = 0D(a + bξ) = b2 Dξ£ 'NP]j\*QQξ ηDξ + Dη~nHUrm*NJIUemD5QyJI\*lYI5JIUrmDn=YIUrmDa, b ∈ IR~ξ ár=â ãrâ const]^bJl§OHN^sLD&WQH]PQHRT_-Nq]j\=bk=D&dHOH_-NqWNj\*QHkHQHOH_n:JI[¾x%ÄSxax Ö ÄSIYDsLDJlz]PWIdH]^JWIz£m*QH]PFHN^GH]PQ=QÁaCE -=ÄSN^dH]^JWQJ.Nj\*lOHIHnom D(ξ + η) = E(ξ + η)2 − (Eξ + Eη)2 == Eξ 2 + 2Eξη + Eη 2 − (Eξ)2 − 2EξEη − (Eη)2 == Eξ 2 − (Eξ)2 + Eη 2 − (Eη)2 = Dξ + Dη.~D(ξ + η) =b!cWdef0g7iFjjÖ *¾ m*NP]Pl"RT_WI]PFHI\*l&stI&W.D\*QH]Pl"]PWIdH]^JWIRzRaDJNPRaDJQHkHNP]tYIUPI'IMzQmD&OHQK T]j\*QM5N]j\=bk=D&dHOH_-NXWNj\*QHkHQHOH_1OHNKHW\=KHf[J]KuOHN^sLD&WQH]PQHRT_-RTQn*RT_¯FHI\=bkHQHR¤^ME><Á'\Ä ¦ÃaCñ+#'!!'1Q\HE><1!!$&%M\'!><'! ξ ' η !;+$+.D(ξ + η) = Dξ + Dη + 2(Eξη − EξEη).Ö (ξ, η) = Eξη − EξEη = E(ξ − Eξ)(η − Eη).SÖ IW.D&GHQ=D&HQK9JD&YMNO=Dst_-W.D&N^J]rKQ!9IE!$& m\ .k"!!$& _ D ! 1]j\=bk=D&dHOH_:VuWNj\*QHkHQHO ξ Q η !$&%,¤BEHEÁ>< Ä1!!¦$&Ã%,aC\'! ><'!Ö U Cq'<E ñ6?"0'<'!%(!='*!><!D }$=)`'ABUB^!!'!'!'U'GCq)+!+H+;+%G$+-+$¥^98F><')+\! .¾x axCE (ξ, η)ρ(ξ, η) = √√ .Dξ DηB'!><'!!$ ξ R ' η !;+$+6^M=`.\'¤E+Á !Ä!$&¦ 7Ã'Ua ClC-\ ' Ö [ (ξ,0|qη)HE=><1!0!'$' ρ(ξ,¾xSa xCE Ö [ s+GHN^pNPOHQK¸sLDmDkHQ η)Ö = [ 0aWQm*OHIHnkJIHn:WIoFHNPGHW_:Vn±LénQnWIoW&JIGH_:VnYIW.D&GHQ=D&HQKD]j\*Njm*IW.DJNj\*lOHInD(ξ + η) = Dξ + Dη + 2è (ξ, η)Q5YI|PZEZEQHHQHNPOJXYIGHGHNj\=KHHQHQ* m*W&b.VOHN^sLD&WQH]PQHRT_:V5]j\=bk=D&dHOH_:V5WNj\*QHkHQHO5W]PNPUrmDG=D&WOKHN^J]KzObL\*fS]%T RT_¨G=D&]P]PRTI&JGHQHR`N^iN[OHNP]PYI\*lYIS]PWIdH]^JWXYIW.D&GHQ=D&HQHQQuYI|PZEZEQHHQHNPOJD9YIGHGHNj\=KHHQHQ¤EÁĦÃaC Ö @ M !D :A!`U^!)=E BHE><1!*1,B'!><'!!$ !.;+$'!><+'!!$ ^]!><';+B$ +EξR^><'!B 9IE!$& N D Rlv/ - k6"A`!U$&^! N)*= k B!HE>< Ñ1!A*1ÜξU^!)..=E6"k !B$& ÛHEξ>< 1!D *1n!B ©'!><A'!`!U$ ^!)ξ=E !E(ξ;+$\−+HE><Eξ)1!^*1><'B\ '!E|ξ|R !9IE!$& N / -*.><'!!$\HEξ><!1!;+$*1+\'!^><'!><!'$ BkRåE=`!'9EI!&$3 D ! A`Uv^)E=E|ξ − Eξ|ξ!;+$+^M><'B Eξ = Eξ(ξ − 1) .
. . (ξ − k + 1) R k¾xSaxCE Ö @ ½cM5N"QsuIFHGHNjm*Nj\*NPOHQKm*QH]PFHNPGH]PQHQ¨WQm*OHIHnTkJIqNPNyOHN]^biNP]^J.W&b.N^JnNP]j\*Q9OHN%]^biNP]^J.W&bNPJRaDJNPRaDJQHkHNP]PYIUPISILM9QmD&OHQKç`m*GbUPId9]^JIokkkk[k]GHIOH_n¦NP]j\*QgQstWNP]^JOHI RaDJNPRaDJQHkHNP]PYINyILM9QmD&OHQHN&n¦OHQHkHNPUPIqI)]^biNP]^J.WIW.D&OHQHQm*QH]PFHNPGH]PQHQ¯]PYDsLDJl§OHNj\*l&sPKÄD&OHOHIN+FHG=D&WQ\*IG=D&]PFHGHI]^JG=D&OKHN^J.]rKêO=D\*fcv&_-NRTIRTNPOJ_}]j\=bk=D&dHOH_:V§WNj\*QHkHQHO§]j\*Njm=bf[iQHR¥IvG=DstIR- T]j\*Q§]^biNP]^J.W&bNPJq\*f%ovId+RTIRTNPOJyFHIG.Km*YD n*JI']^biNP]^J.W&b.N^J'\*fcvId+RTIRTNPOJyFHI&GKm*YD T]j\*Qrq≤r]^biNP]^J.W&b.N^J'RTIRTNPOJ9FHIG.Km*YD nHJIz|^JIHn*WIIv&iNEUPIWIG.Kn=OHNEI&stO=DLk=D&N^Jn*kJI9]^b.oqiNP]^JW&b.N^J"RaI&RaN^OJ9FHIG.Km*YDhr>q¥ÖÖs ê<t!ý Ú ¤:«¥z9²¿Ù w Æß.ÞàHv² Î ª« w ®ß&®H4T -ERT_¨mD\*Q9IFHGHNjm*Nj\*NPOHQK'OHNP]PYI\*lYHQV"O=D&QHvI\*NPNck=D&]^JIW&]^JGHNPk=D&f[iQV]rKG=D&]PFHGHNjm*Nj\*NPOHQHd]j\*Njm=bf[iNtdqJ&D&v\*QHHN5FHGHQHWItm=KJ]rK ]PII&JWN^J]^J.Wbf[iQ=Nz|^JQHRG=D&]PFHGHNjm*Nj\*NPOHQKHRRaDJNPRaDJQHkHNP]PYHQ=NXIM9QmD&OHQK'Q'm*QH]PFHNPGH]PQHQPξEξDξξ ár=â ãrâ aa0Bi(n, p)npnp(1 − p)P ois(λ)λp1−pa+b2λp(1 − p)2(a − b)212a1λσ21λ2G(p)R[a, b]N (a, σ 2 )exp(λ)K(a, σ)S{∞−∞éZ§¨¨ÁË§ë¾ £¾ x axCEÖ t£H:ç:D&RT_-R;FHGHQHRTNPk=DJNj\lOH_-RÀW J&Dv\*QHHN Ö uKHW\=KHN^J]rKI&J]^bJ]^J.WQHNRaDJNPRaDJQHkHNP]PYIUPIILM9QmD&OHQKnTD)]j\*Njm*IW.DJNj\*lOHIHnaQ`m*QH]PFHNPGH]PQHQnTbG=D&]PFHGHNjm*Nj\*NPOHQK)I&pQ aJI9W_stW.D&OHI5JNPR-n=kJIzQHOJNPUPG*DL\pσx1Zdx2π σ + (x − a)2G=D&]eV.ILm*QJ]rK*¦D&YHQHRgIvG=DstIR-nHG=D&]PFHGHNjm*Nj\*NPO=QHNIpQyKHW\=KHN^J]KuItm*OHQHR¨QsXFHGHQoRTNPGHIW§G=D&]PFHGHNjm*Nj\*NPOHQHdnmH\=KYI&JIGH_:VêOHN)]^biNP]^J.W&b.N^J¸RTIRTNPOJIWn-O=DkHQHO=DK]FHNPGHWIUPIH`mD\*lOHNPdpNPR`FHGHQ9GHN^pNPOHQHQzsLDmDkzRT_¨k=D&]^JIXv&bLm*NPR`QH]PFHI\*l&stIW.DJlS]j\*Njm=b.of[iNPNEbJWNPG.M5m*NPOHQHN& ¤Á ux Ö OHNPG=D&WN^OH]^JWIgNPv_p NP W.D. j'ξB @-Ñ= *><!$0[ 1L+`*1O D !RV½vPD)S)@UV^OH6 "/ IP#B εHE><>1!0!*^FA\'!)#><'!'!<P(|ξ − Eξ| ≥ ε) ≤Dξ.ε2(8.9)b!cWdef0g7iFjj[m[Ö ¾ x%ÄSxTx Ö CED&dJQ5m*QH]PFHNPGH]PQHQ']j\=bk=D&dHOH_:V"WNj\*QHkHQHOQ nIFHGHNjm*Nj\*NPOoξ2 ηOH_:VuWsLDmDLk=DV ££5Q+ £¢]PII&JWN^J]^J.WNPOHOHIHÁaCE -*ÁN^pDKysLDmDLkb Ö nHRT_êO=DpX\*QyFHNPGHW_-NERTIRTNPOJ_¯]j\=bk=D&dHOH_:VWNj\*QHkHQHOQ H:I]PFHI\*l&sPbNPRT]Ky]PWIdH]^JWIRêcm*QH]PFHNPGH]PQHQ'mH\=K'NPNW_-kHQH]j\*NPOHQKÄE\=KXO=DVILξM52 m*NPOHηQKSW&JI&GHIUPIcRTIRTNPOJDc]j\=bk=D&dHOHIdSWNj\*QHkHQHOH_WI]PFHI\*l&sPbNPRT]rKSJNPIoGHNPRTId Ö n&FHI\*ILM9QHW.ça\=bk=D&dHO=DK5WNj\*QHkHQHO=DKHW\=KHN^J]rKW_-GHIM5m*NPOog(x) = x2ξ2OHId"WSNjm*QHOHQHHN&=ça\*Njm*IWDJNj\*lOHIHn:_-kHQH]j\*NPOHQHN2− (Eξ2 )2 = 1 − 0 = 1m*QH]PFHNPGH]PQHQYD&YG=DstOHI]^JQ`W&JIGHIUPDξIRT2 I=RTNPEξOJ2 DQYHW.Dm*G=DJDuFHNPGHWIUPIuKHW\=KHN^J]rK`WvI\*l&pQHOH]^JWN]j\=bk=D&NPW9bLm*IvOH_-RQymH\=K)Dv]PI\*f[JOHI"OHNPFHGHNPGH_-WOH_V+]j\=bk=D&dHOH_:VWNj\*QHkHQHORTNPNPRom 22Dη = Eη − (Eη) =[Z10111 14x2 √ dx − = − = .2 x95 945¾ x%ÄSxTx Ö &@ Hça\=bk=D&dHOH_NEWNj\*QHkHQHOH_OHNjsLD&WQH]PQHRT_êQ"ILm*QHO=D&YIoξ1 , .
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