А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 5
Текст из файла (страница 5)
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. . An ) = P(A1 ) · P(A2 | A1 ) · . . . · P(An | A1 . . . An−1 ),(3.3)PN ]j\*Q+W]PNEWVItm=KiQHNSWzFHG=D&W&bfÇk=D&]^Jl"G=D&WNPOH]^JWDzb]j\*IWOH_-NSWNPGHILKJOHI]^JQ+IFHGHNjom*Nj\*NPOH_ÁaCE gsLDmDLkHQy]PI]^JIQJzWOHNPFHI]PGHNjm*]^J.WNPOHOHIR¨FHGHQHRTNPOHNPOHQHQuIFHGHNjm*Nj\*NjoOHQK @ ÄSNPdH]^JWQJN^\*lOHIHnom P(A1 ) · P(A2 | A1 ) · . . . · P(An | A1 . . . An−1 ) == P(A1 ) ·P(A1 A2 ) P(A1 A2 A3 )P(A1 . .
. An−1 An )·· ... ·=P(A1 )P(A1 A2 )P(A1 . . . An−1 )= P(A1 . . . An ).¾ x%ÄSxTx @ s-bGHOH_n.]PItm*NPG.M5DiNPd vNj\*_:V5Q kHNPGHOH_:V5pD&G=DnFHI]eVNPRTN]j\=bk=D&dHOHIUPI9W_-vIG=DvN^sXWI&stWG=DiNPOHQKuFHI]j3\*Njm*IW.DJNj\*2lOHI9QstW\*NPYD&f[J]rKupD&GH_@s ®<th± Ù w Úªv²[«ª« x Úy¥ztÚÙ9zt®+®u«jÝtÞª®Ùr®v¤Ú&Ù9z*³yÙPÚܲWzt®Hànn¦JIUPIHnakJIkHNPGHOH_-d¨pD&G¨WFHNPGHW_-NFHIoC D&dJQ¨WNPGHILKJOHI]^JlEpk k = 1, . .
. , 4KHWQJ]rKuFHGHQ oIRgQH]PFH_J&DOHQ=Qk]PIv_JQHNn5]PI]^JILKiNPNêW1JIR-nkJIÇWÁaCE -z¤vI&stO=DkHQHRËkHNPGHN^sCioR¸QH]PFH_J&D&OHQHQqFHILKHWQJ]rK)kHNPGHOH_-d+pD&Gb]^JlWFHNPGHW_-NkHNPGHOH_-d+pD&GiFHILKHWQJ]rKFHGHQ oIRgQH]PFH_JD&OHQ=Q %GHNPv&b.NjJ]KO=D&BdJk Q = {ç:Iv_JQKk}pk = P(Bk )BkRTILM9OHI5W_-G=DstQJlzkHNPGHN^sQ]j\*Njm=bf[iQ=R¸IvG=DstIR-hom CiB1 = C 1 ,CiB 2 = C 1 C2 ,B 3 = C 1 C 2 C3 ,B 4 = C 1 C 2 C 3 C4 .I5ZEIGHR¦bt\*N" @ @ :FHI\=bk=D&NPRP(B1 ) = P(C1 ),P(B2 ) = P(C 1 )P(C2 | C 1 ),P(B3 ) = P(C 1 )P(C 2 | C 1 )P(C3 | C 1 C 2 ),P(B4 ) = P(C 1 )P(C 2 | C 1 )P(C 3 | C 1 C 2 )P(C4 | C 1 C 2 C 3 ).I5Y\D&]P]PQHkHNP]PYIR¦buIFHGHNjm*Nj\*NPOHQHf} @ W&NPGHIKJOHI]^JQQHRTNPNPR2P(C1 ) = ,53P(C 1 ) = ,5P(C i+1 | C 1 .
. . C i ) =3−i,5−i2, i = 1, 2, 3.5−iGHN^sPbL\*lLJ&DJNSFHI\=bk=D&NPRnnQp1 = 0.4 p2 = 0.3 p3 = 0.2 p4 = 0.1P(Ci+1 | C 1 . . . C i ) =¾x%ÄSxax @ Ô&Á¦DstGH_-W|j\*NPYJGHQ=kHNP]tYIdzHNPFHQ5FHGHIQH]eV.ILm*QJXWcJIR ]j\=bk=D&N&nYIoUemDW_:V.ILm*QJzQsE]^JGHILK'V.I&JLKuv_êItm*QHOyQsJGHNjVyFHI]j\*Njm*IW.DJNj\*lOHIz]PINjm*QHOHNPOHOH_V|j\*NPRTNPOJIWH¤FHGHNjm*Nj\*QJlWNPGHILKJOHI]^JluJIUPIHnkJIuOHN5v&bLm*N^JuG=DstGH_-W.DyHNPFHQnNP]jo\*Q |j\*NPRTNPOJ_;W_:V.Itm=KJQsz]^JGHIK OHN^sLD&WQH]PQHRTIym*GbU5I&J'm*GbU^Dy]PII&JWN^J]^J.WNPOHOHI]WNPGHILKJOHI]^JLKHRTQnQED&Y)QstRTNPOHQJ]rK QH]PYIRaDK)WNPGHIKJOHI]jJlHnNP]j\*Q0.3 0.4 0.6QstWNP]^JOHIHn*kJIzFHNPGHW_-d|j\*NPRTNPOJ"OHNEW_-dm*N^J9QsX]^JGHILK*wÁaCE -%]PYIRaDKWNPGHILKJOHI]^JlG=D&WO=D`WNPGHILKJOHI]^JQJIUPIHnkJI§OHNpW_-dm=bJcQsa]^JGHIKXW]PN¦JGHQS|j\*NPRTNPOJD&¤vI&stO=DLkHQHRodE|j\*NPRTNPOJcOHNaW_-dm*N^Jk = {kQsc]^JGHILK n.IUrmD¦AD&YzYD&Y9]PIv_JQKKHW\=KHf[J]rK} k = 1, 2, 3p = P(A1 A2 A3 )AiOHN^sLD&WQH]PQHRT_-RTQn*QHRTNPNPRom T]j\*QuQstWNP]^JOHIHnkJIzFHNPGHW_-d|j\*N^RaN^OJ"OHNEW_-dm*N^J9QsS]jJGHIKnHJIp = P(A1 )P(A2 )P(A3 ) = (1 − 0.3) · (1 − 0.4) · (1 − 0.6) = 0.168.p = P(A1 A2 A3 | A1 ) = P(A2 )P(A3 ) = 0.24.!b cWde(fhgiK®M5~_7 77@ ¾x%ÄSxTx @ Ö ÁN^pQJ.l)sLDmDLkb @ nNP]j\*QV.I&JLKv_ÀItm*O=DQs'WNPGHILKJOHI]^JNPdQ\*QG=D&WOKHN^J]rKObt\*fSP(A)P(B)¾x%ÄSxTx @ jÄSIYDsLDJl%]PWIdH]^JW.D%OHN^sLD&WQH]PQHRT_:VE]PIv_JQHdnFHI\*l&sPbKH]Pl%IFHGHNjom*Nj\*NPOHQHNPR WNPGHIKJOHI]^JQQOHN^sLD&WQH]PQHRT_ÀW]PIWIYbFHOHI]^JQn¾x%ÄSxTx @ b.]^Jl+]PIv_JQK nA B CFHGHQHkHNPR1YDM5m*INQsz|^JQV]PIv_JQHdQHRTNPN^JWNPGHILKJOHI]^JlHnI&Jt\*QHkHObfäI&JuObL\=K QNjm*QHOHQHH_=uIU^bJS\*Q9]PIv_JQKnQv_JlFHIF=D&GHOHISOHN^sLD&WQH]PQHRT_-RTQ'QAB BC ACOHN^sLD&WQH]PQHRT_-RTQuW5]PIWI&YbFHOHI]^JQ*wstRTNPOHQJ]rK'\*QyI&JWN^Jn=NP]j\*Q"V.I&JKyv_ILm*OHIQs]PIv_JQHd n QQHRTNPN^J9WNPGHILKJOHI]^Jl9ObL\*lzQ\*QuNjm*QHOHQH=D.wA B C¾x%ÄSxTx @ 1Njm*QHOHQHkHOH_-d YHW.Dm*G=DJ'O=DLbtmDLkb+vGHI&pNPO=D"JIkHYDED&YIW.DWNPGHIKJOHI]jJl9JIUPIHn*kJIzJIkHYDzv&btm*N^J9bLmD\*NPO=DzI&JzHNPOJG=D9YHW.Dm*G=DJDzO=D5G=D&]P]^JIon*NP]\*Q+QstWNP]^JOHIHnkJI"I&J9YDM5m*IduQsS]^JIGHIO+YHW.Dm*G=DJDKHOHQHNXRTNPOHl&pNPN&nkHNPR1/3IO=DbtmD\*NPO=DzvI\*l&pN&n*kHN^R¸O=Dw1/6¾x%ÄSxTx @ b]^Jl5]PIv_:JQKQOHN^sLD&WQH]PQHRT_QQOHN^sLD&WQH]PQHRT_A BA CIYDstDJlHn=kJIQRTIU^bJ9OHNEv_JlzOHN^sLD&WQH]PQHRT_-RTQA B∪C¾x%ÄSxTx @ @ CEDdJQ9WNPGHILKJOHI]^JlXJIUPIHn.kJIXFHGHQzvGHI]tD&OHQHQ5JGHNjVzQHUPG=D\*loOH_:VYI]^JNPd V.I&JLKv_»O=DyItm*OHId W_-F=Dm*N^JpNP]^Jl+IkHYIWHnFHGHQ b]j\*IWQHQnkJIuO=DW]PNjVuYI]^JLKVuW_-F=Dm=bJ9UPG=D&OHQu]EkHN^JOH_-RkHQH]j\*IR¨IkHYIWHnQ¾x%ÄSxTx @ t£Hça\*Njm=bN^Ju\*QqFHIF=D&GHO=DKqOHN^sLD&WQH]PQHRTI]^Jlu]PIv_JQHdA1 A2QsXG=D&WNPOH]^JW.DwA3P(A1 A2 A3 ) = P(A1 )P(A2 )P(A3 )¾x%ÄSxTx @ ¢Hb]^Jl{äOHN^sLD&WQH]PQHRT_-NEW]PIWIYbFHOHI]^JQu]PIv_oA1 , .
. . , A nJQKHÄSIYDsLDJlHn=kJIQP (∪ni=1 Ai ) = 1 − ni=1 P(Ai ).¾x%ÄSxTx @ L*:NPGHOHI\*QyG=D&WNPOH]^JWIwP(A| B) + P(A| B) = 1¾x%ÄSxTx @ b]^JlnQ:NPGHOHIP(A| B) > P(B| A) P(A) > 0 P(B) > 0\*Qn=kJIwP(A) > P(B)¾x%ÄSxTx @ Ö CEDLbtmDkbEvGHI&pNPOHI[m*WNQHUPG=D\*lOH_-N:YI]^JQ&CED&dJQSb]j\*IWObfWNPGHIKJOHI]jJl`JIUPIHnakJIW_-F=Dm=bJ m*WN'FKJNPGHYHQnNP]\*Q¸QstWNP]^JOHIHn:kJI]^bRTRaDW_-F=D&W&pQVuIkHYIWm*Nj\*QJ]KO=DFKHJl¾x%ÄSxTx @ LISHNj\*Q"FHGHIQstWItm*QJ]K OHN^sLD&WQH]PQHRT_:V"W_-]^JGHNj\*IWH=:N^GHIoKJOHI]^JlyFHIF=DmD&OHQK+FHGHQ oIR¸W_-]^JGHNj\*N5G=D&WnO=D nCEDdJQ)WNPGHIoipi i = 1, . . .
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-+mDOHOHIdEsLDmDLkHNaOHNPWI&sPRaILM9OHI-IFHGHNjm*Nj\*QJlD&FHGHQHIGHQnL]PYI\*lYIvG=D&YIWD&OHOH_:VyQs^m*Nj\*QHd'm*Iv.D&W\*NPOHIWKiQHY=GHQy|^JIR§\*NPUPYIzFHNPGHNPkHQH]j\*QJ.lzW]PNWI&stRTIMzOH_-NW.D&GHQ=D&OJ_¥W_-vIG=D'Qs^m*Nj\*QHd ]PIy]PY\DmDhRT_¿RTIMNPRW&sPKJlyW'JIkoOHI]^JQObt\*lHnILm*OHIQ\*Qqm*W.DuvG=D&YIW.DOHOH_:VQs^m*Nj\*QK{ÑQHOHIUPIOHN5mD&OHIH¦D&YHQHRIvG=DstIR-nRT_`G=DstvQHW.D&NPR+W]PN:FHGHI]^JG=D&OH]^JWI|j\*NPRTNPOJD&GHOH_VQH]rV.Itm*IWO=D%FHI\*ObfQsXm*W&b.VQs^m*Nj\*QHdnW&sPKJ_:V+]PI9]PY\DmDnGHIWOHI v&bLm=bJUPGbFHFb]PIv_JQHdBk = {kvG=D&YIWD&OHOH_-RTQ nnFHI]PYI\*lYbQn} k = 0, 1, 2B0 ∪ B1 ∪ B2 = Ω Bk Bl = ∅ k 6= l:I&stOHQHYHOHIWNPOHQHNFHI\*OHIdUPGbFHFH_Ç]PIv_JQHd+W9ZEIGHR¦bt\*QHGHIWYNXsLDmDLkHQuKHW\=KHN^J]rKWNPGHOH_-RêFHGHQstO=D&YIRJIUPIHnkJI'FHGHQ)GHN^pNPOHQHQ O=D&RFHGHQm*N^J]rK WI]PFHI\*l&stIW.DJl]rKZEIGHR¦bL\*IdSFHI\*OHIdSWNPGHILKJOHI]^JQQ\*QZEIGHR¦bt\*Id D&dHNP]tDb.]^JlQs^m*Nj\*QHN&nW&sPKJINQsFHIFHI\*OHNPOHOHIUPI"KiQHYDnv&bLm*N^JyOHNvG=D&YIW.D&OHOH_-R 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kHNPGHOH_:VpD&G=DnFHIE]eV.NPRTNW_-vIG=D9vN^s5WI&stWG=DiNPOHQK)I&JIvG=D\*QpD&4G=D »D&Gn2W&sPKJ_-dqO=DtbLmDLkb+Qs|^JQVm*W&b.Vn=IYDsLD\*]rKuvNj\*_-R-*ED&YIW.DWNPGHIK2JOHI]^Jl5JIUPIHn=kJIzW&JIGHIdypD&G'JIMNvNjo\*_-d*wÁaCE -¾*m*NP]PlHn=YD&YuQyWzFHGHNjm*_:m=biNtdusLDmDkHNn*RT_¯QHRTNPNPR§m*Nj\*Iz]EFHI\Ho]PGHNjm*Q"m*W&b.V'I&JIvG=D&OHOH_:V'QsbGHOH_pD&GHIWGHIWOHIOHId'UPGbFHFHIdy]PIv_JQHdvNj\*_:V nBkb]^=Jl {pD&GnW&sPKJ_-dqO=DtbLmDLkb+QsSm*W&b.V+I&JIvG=D&Ook} k = 0, 1, 2A={OH_:VnLvNj\*_-d L%GHNPv&bN^J]rKSO=D&dJQEb.]j\*I&WObfgWNPGHILKJOHI]^JlI-ZEIGHR¦bL\*N}P(B2 | A) D&dHNP]tD5QHRTNPNPR[{oom P(B2 | A) =P(B2 )P(A| B2 ).P(B0 )P(A| B0 ) + P(B1 )P(A| B1 ) + P(B2 )P(A| B2 )I\*l&sPb&KH]PlcG=D&]P]^bMm*NPOHQKHRaQnFHGHQHWNjm*NPOHOH_-RTQW[GHN^pNPOHQHQSsLDmDkHQzÔ¢nLFHI\=bk=D&NPRP(B0 ) =C221= ,2C615P(B1 ) =8C41 C21= ,2C615P(B2 ) =6C42= .2C615sm*Wb.VuI&JIvG=D&OHOH_:VypD&GHIW5vNj\*_-dupD&GuRTILM9OHI5QsPW\*NPkHl9]EWNPGHILKJOHI]^JLKHRTQP(A| B0 ) = 0,P(A| B1 ) = 0.5,P(A| B2 ) = 1.ça\*Njm*IW.DJNj\*lOHIHnP(B2 | A) = 0.6¾x%ÄSxax¼£H @ ¦Nj\*NPUPG=D&ZEOHINu]PIIv&iNPOHQHNu]PI]^JIQJ`Qsu]PQHUPO=D\*IWqºJIkHYDyQºJQHGHNj:çJ&DJQH]PJ.QHkHNP]tYHQHN+]PWIdH]^JW.D FHIRTNjVgJD&YIW_nakJI`QH]PYDM5D&f[J]rK¸W ]PGHNjmHoOHNPR]PIIv&iNPOHQHdºJIkHYD9Q]PIIv&iNPOHQHdºJQHGHN¦stWNP]^JOHIHnkJI+]PGHNjm*QFHNPGHNjm2/5D&W.D&NPRT_:V`]PQHUPO=D\*IWyºJIkHYD1/3zQqºJQHGHNj9W]^JGHNPk=D&f%J.]K`WI&JOHI&pNPOHQHQ5:3¤FHGHNjm*Nj\*QJl5WNPGHILKJOHI]^JlJIUPIHnkJIv&btm*N^J5FHGHQHOKJFHNPGHNjmD&W.D&NPRT_-d']PQHUPO=D\nNP]jo\*QD. :v&bLm*N^J9FHGHQHOKJz]PQHUPO=D\ºJIkHYD~Hv v&btm*N^J9FHGHQHOKJ9]PQHUPO=D\ºJQHGHNjv&btm*N^JFHGHQHOKJ]PQHUPO=D\ºJIkHYD Qv&bLm*N^JÁaCE -b]^JlA={} B ={FHGHQHOKJ+]PQHUPO=D\`ºJQHGHNj ¦ç:Iv_JQKv&bLm*NjJ)FHNPGHNjmD&O`]PQHUPO=D\ºJIkHYD Qv&bLm*N^JSFHNPGHNjmD&O"]P}QHUPO=D\9ºJQHGHNj C1I=vG=D{sPbf[JFHI\*ObfêUPGbFHFbIEb]j\*IW}QHfC2 = {n]j\*Njm*IW.DJNj\*lOHIH}nQI]PYI\*lYbom P(C1 ) : P(C2 ) = 5 : 3P(C1 ) = 5/83P(A| C1 ) = ,5P(C2 ) = 3/81P(A| C2 ) = ,3b!cWde(fhgiKÀ2P(B| C1 ) = ,5FHIzZEIGHR¦bt\*NS D&dHNP]tD5QHRTNPNPRP(C1 | A) =P(C2 | B) =@ 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