А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 6
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. . , Aδnn =nYi=1P Aδi i(5.1)mH\=KW]PNjV5O=DvIGHIWnnn&Urm*N-]PIv_JQHN δi NP]^Jlºb]PFHNjVW oIRuQH]PFH_(δJ&1D&,OH.QH. Q. ,nNPδ]n\*)Q δi ∈ {0,ntQ1}OHN^ib=]PFH1,NjV. n. NP. ]j, \*nQPÄSGbUPQHARTiQS]j\*Ioiδi = 1δi = 0W.D&RTQnHmH\=Ky\*fcvId+HNPFHI&kHYHQmH\*QHOH_ n]PI]jJIKiNPd)QsXºb]PFHNjV.IWSQyOHN^b.]PFHNjV.IWnWNPGHILKJOHI]^Jl"FHGHIQstWNjm*NPOHQK+|^JQV]PInv_JQHd+G=D&WOKHN^J]KFHGHIQstWNjm*NPOHQHf»W&NPGHIKJoOHI]^JNPd']PIv_JQHdnJINP]^JlSmD&OHO=DK9HNPFHIkHYDS]PI]^JIQJQsOHN^sLD&WQH]PQHRT_:V"]PIv_JQHdÃXNPUPYIFHIYDsLDJlHnkJIW_-FHI\*OHNPOHQHNEG=D&WNPOH]^JW.Dy¢ aW\*NPkHN^JzOHN^sLD&WQH]PQHRTI]^Jl5]PIov_JQHd δ1Wc]PIWIYbFHOHI]^JQ&I]^JGHINPOHObf¸RTItm*Nj\*lcO=Dst_-W.D&f[J]rV.NPRTIdA1 , .
. . , AδnnOHN^sLD&WQH]PQHRT_:VuQH]PFH_J&D&OHQHd+ :NPGHObL\H\*Qb]^JlXFHGHI]^JG=D&OH]^JWIX|j\*NPRTNPOJD&GHO=_:Vz]PIv_:JQHdIFHQH]P_-W.D&N^JE]rV.NPR¦b QH]PFH_:oJD&OHQHd :NPGHObt\H\*QnJIyNP]^JluYDM5m*IR¦b|j\*NPRTNPOJD&GHO=IΩR¦b)QH]eVItm=b]^J&nD&WQJ]rKω∈ΩWIW&sLD&QHRTOHIEILm*OHI&stO=DLkHOHIN%]PII&JWN^J]^JWQHNcOHN^YI&JIG=DKzHNPFHIkHYD δ1Q\*QnA1 , . . . , AδnnkJIJIMN]tD&RTIN&nOHNPYI&JIGH_-d'm*WIQHkHOH_-d'O=D&vIG=¤]POHIW_-W.DKH]PlO=D(δ1 , .
. . , δn )@Öb!cWde(fhgi]Í@ =G D&WNPOH]^JWNz¢ jnIFHGHNjm*Nj\*QHR§O=DSYDM5m*IRItm*OHI&JIkHNPkHOHIR§RTOHILMNP]^JWNWNPGHIo{ω}KJOHI]^Jl9]j\*Njm=bf[iQHR¸IvG=DstIR-hP({ω}) = pk (1 − p)n−k ,(5.2)Urm*NNP]^Jl)YI\*QHkHNP]^JWIºb]PFHNjV.IW9WQH]PFH_JD&OHQKV§ :NPGHObL\H\*QQ\*Qnm*GbUPQHRTQj] \*IW.kD&RTQnYI\*QHkHNP]^JWINjm*QHOHQH`WuO=D&vInGHNb.]^Jl]j\=bk=D&dHO=DK* WNjo(δ1 , . . . , δn )\*QHkHQHO=DG=D&WOKHN^J]rK+YI\*QHkHNP]^JW&byºb]PFHNjVIWXWQH]PFH_JD&OHQKVIUrmDmH\=K+WNjoξnnGHIKJOHI]^JQFHILKHW\*NPOHQKyGHIWOHI ºb.]PFHNjV.IWW QH]PFH_JD&OHQKV :NPGHObL\H\*Qkn]PFHG=D&WNjmH\*QHWPI9(n,G=DWk)NPOH]^JWIXP (n, k) = P(ξn = k) =ω: δ1 +...+δn =kP({ω}) = Cnk pk (1 − p)n−k ,(5.3)FHI]PYI\*l&YbkHQH]j\*Ig]PIkHN^JD&OHQHdk NP]^Jl¨YI\*QHkHNP]^JWIgW]PNjV¸m*WIQHkHOH_:VO=D&vIGHIWmH\*QHOH_n¦]PItm*NPG.M5DiQVgW+JIkHOHCIn]^JQNjm*QHOHQHaED&Y§RT_äbW&Qm*QHR¥W 9n¦kHQH]jonkIFHGHNjm*Nj\=KHf[JqJD&Y§O=Dst_-W.D&NPRTINQ\*Q\DP (n, k)q]j\=bk=D&dHOHIdWNj\*QHkHQHOH_¦¾D&RTN^JQHR-nkJIm*IWI\*lOHIk=D&]^JIcFHItmG=D&]PFHGHNjm*Nj\*NPOHQHNtR) :NPGHObL\H\*QSFHILm*G=DsPbRTNPW.ξD&nf[JcvQHOHIRTQ=D\*lOHINaG=D&]PFHGHNjom*Nj\*NPOHQHNSFHGHQn=1FHIKHW\*NPOHQK)OHNRTNPOHNPN ºb.]PFHNjo¾x%ÄSxTxÀ¢ CED&dJQqWNPGHILKJOHI]^JlR(n, k)VIWW5]eVNPRTN OHN^sLD&WQH]PQHRT_:VQH]PFH_JD&OHQHd) :NPGHObt\H\*Q§ j knk = 0, .
. . , nÁaCE -*¤kHNPWQm*OHIHnkJI5QH]PYIRaDK'WNPGHILKJOHI]^Jl5W_-kHQH]j\=KHN^J]KyFHIZEIGoR¦bL\*NA)+\!'(Ä@B!H'LT/ '!D 7'!9IEA)+B!'om R(n, k) =nXi=kP (n, i) = 1 −k−1XP (n, i).(5.4)i=0¤]PIv_-d5QHOJNPGHNP]cFHGHNjm*]^JD&W\=KHN^JXstO=DkHNPOHQHN%WNPGHILKJOHI]^JQFHILKHW\*NPOHQKV.IoJK9v_¸Itm*OHIUPIEºb]PFHNjVD[W QH]PFH_JD&OHQKVnYI&JIG=DK9W_-kHQHR(n,]j\=KHN^J.1)]rK'kHNPGHN^sWNPGHILKJoOHI]^Jl5m*IFHI\*OHQJNj\*lOHIUPI9n]PIv_JQKuFHIzZEIGHR¦bt\*N[R(n, 1) = 1 − (1 − p)n .¾x%ÄSxTx ¢ [JI¨WNPGHILKJOHNPNW_-QHUPG=DJl¸bG=D&WOHI]PQ\*lOHIUPI¨FHGHI&JQHWOHQHYD OHQHkHNPdHOH_-dQH]eV.ILm§F=DGJQHQQH]PY\*fckHNPO* jhD. JGHQ`F=D&GJQHQ`Qs9kHN^J_-GHNjVQ\*QFKJlF=D&GJQHd+QsXWI]PlRTQ~*v :OHNXRTNPOHNPNXJGHNjVF=DGJQHd+QsSkHN^J_-GHNjV+Q\*QuOHNXRTNPOHNPNXFKHJQF=D&GJQHdQsXWI]PlRTQ*wÁaCE -=¦D&Y'YD&YyFHGHI&JQHWOHQHYHQuG=D&WOHI]PQ\*lOH_-N&nHJI5WNPGHIKJOHI]^Jlºb]PFHNjoVD"W_-QHUPGH_pD. [W9YDM5m*IR¸QH]PFH_JD&OHQHQ¨F=D&GJQHQ* [G=D&WO=D=ÄE\=KuJIUPIHnkJIo1/2v_¼I&JWN^JQJl§O=DWIFHGHI]P_ sLDmDLkHQnam*I]^JDJIkHOHI§O=D&dJQWNPGHILKJOHI]^JQnP (4, 3)nQ=I5ZEIGHR¦bL\D&R¢ @ :Q¢ £ :FHI\=bk=D&NPRom P (8, 5) R(4, 3)R(8, 5)P (4, 3) = C431711= >= C83 8 = P (8, 3),424322s Ít ×Î «D¤ÞÈ%« x <Æ wvw ®£ R(4, 3) =4XP (4, i) =i=38X593P (8, i) = R(8, 5).<=16256 i=5¾ x%ÄSxax¢ @ &`F=D&GJQHQzQsQs^m*Nj\*QHd5YDM5m*I&NQs^m*Nj\*QHN%OHN^sLD&WQH]PQHRTIn = 200I&J+I&]^J&D\*lOH_:V`RTILMN^Jv_Jl+vG=D&YIW.D&OHOH_-R]zWNPGHILKJOHI]^JlfCED&dJQWNPGHILKJOHI]^JlJIUPIHn:kJI§kHQH]j\*IvG=D&YIW.D&OHOH_:V¸Qs^m*Nj\*QHdW`|^JIpd=F=D&0.01GJQHQv&btm*N^JG=D&WOHIJGHNPR-ÁaCE - bLm*NPR¨FHIOHQHRaDJl5FHItmyºb.]PFHNjV.IRE]PIv_JQHN&n=]PI]^JIKiNPNXWJIR-nkJIuQs^m*Nj\*QHN5KHW\=KHN^J]K vG=D&YIW.D&OHOH_-R-]PFH_JD&OHQHN" :NPGHObt\H\*Q`]PI]^JIQJWyFHGHIoWNPGHYN9Qs^m*Nj\*QKO=DuvG=D&Y%GHNPv&bN^J]rKO=D&dJQ`WNPGHIKJOHI]^JlIZEIGoP (200, 3)R¦bt\*N"¢ @ :QHRTNPNPRom 3P (200, 3) = C200(0.01)3 (0.99)197 .:_-kHQH]j\*QJl)WGbkHObfÂFHG=D&W&bf}k=D&]^Jl+FHI]j\*Njm*OHNPUPI)G=D&WNPOH]^JW.D+FHGHNjm*]^JD&W\=KHN^J.]rKm*I]^JDJIkHOHIJGbLm*OHId"sLDmDLkHNPdH¤[m*O=D&YIHnQHRTN^K9YD\*lYbt\=KJIGnRTILM9OHIXFHI\=bkHQJlI&JWN^JHhP (200, 3) ≈ 0.18136¾x%ÄSxax»¢ £HêF=D&GJQHQ)QsQs^m*Nj\*QHd)YDM5m*INXQsjm*Nj\*QHNOHN^sLD&W&Qon = 22500]PQHRTI)I&J+m*GbUPQV§RTILM5N^J)v_JlqvG=D&YIW.D&OHOH_-RÇ]'WNPGHILKJOHI]^Jlf¦CED&dop = 0.2JQWNPGHIKJOHI]jJl9JIUPIHnkJI9kHQH]j\*IvG=D&YIW.D&OHOH_:VuQs^m*Nj\*QHd+v&bLm*N^J9sLD&Y\*fckHNPOHIξnRTN^M5m=bQ4380 4560ÁaCE -H¦D&Y9MN&nYD&Y"Q'WSsLDmDLkHNE¢ @ nv&bLm*NPR§FHIOHQHRaDJlFHItm'ºb]PFHNjV.IR]PIv_JQHN&n*]PI]^JIKiNPNXW5JIR-n=kJI9Qs^m*Nj\*QHNEKHW\=KHN^J]KuvG=D&YIW.D&OHOH_-R-¤kHNPWQm*OHIHnkJIzQH]PYIRaDKuWNPGHILKJOHI]^JlO=DV.ILm*QJ]rKFHI5ZEIGHR¦bL\*Nom PP =4560XP (22500, k) =4560XkC22500(0.2)k (0.8)22500−k .k=4380k=4380ÄE\=KEJIUPIHnkJIv_ m*IWNP]^JQXm*IckHQH]j\D[FHG=D&W&bfgk=D&]^JlFHI]j\*Njm*OHNPUPIG=D&WNPOH]^JW.DnbMNOHNm*I]^JDJIkHOHISRTI&iOHIUPIXYD\*lYbt\=KJIG=D{¿JGHNPv&bN^J.]rK9O=D&FHQH]tDJlSFHGHIUPG=D&RTR¦bO=DYIRTFHlf[JNPGHN&ED&Y`RT_ÀWQm*QHR-nO=DVILM5m*NPOHQHN"WNPGHILKJOHI]^JNPdRaILMN^J+v_Jl)]PIFHGKoP (n, k)MNPOHI ]uvI\*l&pQHRTQ¨W_-kHQH]j\*QJNj\*lOH_%RTQ¸]j\*IM9OHI]^JLKHRTQ:¤[m*O=D&YIHna]^biNP]^J.W&bf[J]PFHI]PIv_æv_-]^JGHIgQm*I]^JDJIkHOHIgJIkHOHI|^JQêWNPGHILKJOHI]^JQcuN^JItm*_JD&YHQVuIHNPOHIY'mD&f[Jm*WNEOHQMNP]j\*Njm=bf[iQHNXJNPIGHNPRT_ÄSIIFHGHNjm*Nj\*QHR¸kHQH]j\D]j\*Njm=bf[iQHRIvG=DstIR-hF*!'!"IP (n, k)P (n, k) =(Cnk pk (1 − p)n−k0FHGHQFHGHQ k = 0, .
. . , n;k = n + 1, n + 2, . . . . GHNjm*FHI\*ILM9QHR-nTkJIWNPGHIKJOHI]^Jlqºb.]PFHNjVDuW Itm*OHIR¥QH]PFH_J&D&OHQHQgmH\=K¨]eV.NPRT_n]PI]^JILKiNPd QsQH]PFH_JD&OHQHdnsLD&WQH]PQJyI&J nQ)IvI&stO=DLkHQHRêNPNkHNPGHN^sç:FHG=DonnpnWNjmH\*QHW.D5]j\*Nm=bf[i5DKuJNPIGHNPRaDb!cWde(fhgi]Í£H@¤Áux¢ + JNPIGHNPRaDb.D&]P]PIO=D. jh)A98F'< C><A'p −→ 0n → ∞\'< 7IEA!¾*""><''!\==EC +><RG½! PDp)n )−→U^:λ-% Ak' =n →0, 1,∞.C.
.PD)+A 'λ n2Ï→=∞*'<` -SÐA --8F'! .nnPD)+><'B π *A)+\U^!6^-P*"(n,*k)k−→!'\ π ,kkπk = e−λSλk.k!(5.5)¾x axCE ¿¢ GHQ`FHG=D&YJQHkHNP]tYIR¿FHGHQHRTNPOHNPOHQHQJNPIGHNPRT_¢ "FHGHNjmHoHF I\D&U^Df[Jn:kJI{ ZEQHY]PQHGHIW.D&OHO=DK¸WNPGHILKJOHI]^Jl`ºb]PFHNjVD+W]eVNPRTN+]pn = p¨|jJIR§]j\=b.osLDmD&OHOH_-R§kHQH]\*IR§QH]PFH_JD&OHQHd nkHQH]j\*I FHI\D&U^D&f[JG=D&WOH_-Rnλnpk=D&NE]PFHG=D&WNjmH\*QHW.DzD&FHFHGHIY]PQHRaD&HQKP (n, k) ≈ πk .Ñ;+=T+*`D ¯v&btG=m*D&N^J¨]PFHFHGHINjm*YNjD\*sLD&NPOHO=QDK]PW&WKNPGHstl§IKkHJOHQHI]PNj]^\JNPdπ ]+JD&YêO=Dst_-W.D&NPRT_-R©A!HE--*=E'< ¾xSa xCE§ ¢ [ NPIGHNPRaDXb.D&]P]PIO=DmD&N^JXV.IGHI&pNPNcFHGHQHv\*QM5NPOHQHN&nNP]j\*QWNj\*QHkHQHO=D m*I]^JDJIkHOHIRaD\DnI&JOHI]PQJNj\lOHIIv&iNPUPIkHQH]j\DcQH]PFH_JD&OHQHd u_kv&bLm*NPR¨]PkHQpJDJ.lHn*kJIz|^JI5b.]j\*IWQHNEW_-FHI\*OHNPOHIHn*NP]j\*Qnp2 < 0.1 ¤Á uxæ¢ QHOJNPUPG=D\*lO=DKgJNPIGHNPRaDybD&WG=DoºÃSD&F\D&]tD ^n@ C 2 =`'!>U[-7HA%V 'A!$0"!'<^<%Ä@B!HE'R½v5PDV)HA"'I p =const ξnn→<' 3 <'7 !A*"!'E=∞nξn − npx21 Z − u2P x1 ≤ q≤ x2 −→ √e 2 du2π x1np(1 − p)(5.6)+D BA x ' x Ò −∞ ≤ x ≤ x ≤ +∞Ó R¾xSa xCE;¢ @ GHQgW_-kHQH]j\*NPOHQHQFHG=D&WIdgk=D&]^JQ¸]PII&JOHI&pNPOHQKÇ¢ . QH]PFHI\*l&sPbf[J.]rKZbOHYHHQHQ121x1 Z − u2e 2 duΦ(x) = √2π −∞Q2x1 Z − u2e 2 du,Φ0 (x) = √2π 0ZT:`E 79IE!$& Z;+=* stO=DkHNPOHQK YI&JIGH_:V FHGHQqsLDmD&OHOH_:VRTIM9OHI'O=D&dJQ W'J&D&v\*QH=DV S"v&bLm*N^JxFHIYDsLDO=Dz]PW&KstlmD&OHOH_:VZbOHYHHQHd+]EJD&YO=Dst_-W.D&NPRT_-RG=D&]PFHGHNjm*Nj\*NPO=QKWNPGHILKJOHI]^JNPds Ít ×Î «D¤ÞÈ%« x <Æ wvw ®¾xSaxCE;¢ £H¦NPIGHNPRaD ¢ [ mD&N^J)V.IGHI&pNPNuFHGHQHv\*QMNPOHQHNuWq]j\=bk=D&NnYIUemDyWNj\*QHkHQHO=Dm*I]^JDJIkHOHIWNj\*QHYDu_¥v&btm*NPR1]PkHQJDJlHnkJI|^JI[£b]j\*IWQHNXW_-FHI\*OHNPnp(1OHIHn=NP−]j\*Qp)np(1 − p) > 20ÄE\=K JIUPIHnkJIv_¥bvNjm*QJl]K`W'm*NPdH]^JWNPOHO=I]^J.Q`FHGHQHRTNPOHNPOHQKJNPIGHNPR¥¢ zQ¢ nHGHN^pQHR¨sLDmDLkHQ+¢ @ Q+¢ £HnHOHN^stO=DLkHQJN^\*lOHI9QstRTNPOHQHW9QV'b]j\*IWQKQs^m*Nj\*QHd5YDM5m*I&NQs^m*Nj\*QHN%OHN^sLD&WQH]PQHRTI¾x%ÄSxax¢Ô¢&`F=D&GJQHQzQsI&JI]^JD\*lOH_:V5RTILMN^Jv_JlvG=D&YnI=W.D&OH200OH_-Rq]WNPGHILKJOHI]^Jlfp = 0.01WNPGHILKJOHI]^JlJIUPIHn:kJI§kHQH]j\*IvG=D&YIW.D&OHOH_:V¸Qs^m*Nj\*QHdW`|^JIdF=D&GJQHQv&btm*N^JG=D&WOHIJGHNPR-]EFHIRTI&ilfJNPIGHNPRT_ÇbD&]joÁaCE -¤HNPOHQHR¸WNPGHILKJOHI]^JlP (200, 3)]PIO=DÄSNPdH]^JWQJNj\lOHInJNPIGHNPRaDq¢ 5mD&N^JVI&GHI&pNPN'FHGHQHv\*QMNPOHQHN&nFHI]PYI\*lYbWNj\*QHkHQHO=Dm*I]^JDJIkHOHI9RaD\D=RTNPNPRn2[{#!'!"Iom np = 0.02λ = np = 2P (200, 3) ≈ e−223≈ 0.1805.3!T]j\*Qg]PG=D&WOHQJl FHI\=bkHNPOHOH_-d¸GHN^sPbt\*lLJ&DJ`]'I&JWN^JIR¥sLDmDLkHQ¨¢ @ n¦RT_äbWQm*QHR-nkJIzI&JOHI]PQJNj\*lO=DKFHIUPGHN^pOHI]^J.l9]PI]^JD&W\=KHN^J"W]PNPUPI Ô¢ "¾x%ÄSxax»¢ êF=D&GJQHQ)QsQs^m*Nj\*QHd)YDM5m*INXQsjm*Nj\*QHNOHN^sLD&W&Qo]PQHRTIqI&J+m*GbUPQV§RTILMN^Jqv_JlqvnG=D&=YI22500W.DOHOH_-R¿]'WNPGHILKJOHI]^Jlfp = 0.25WNPGHILKJOHI]^JluJIUPIHnkJIukHQH]j\*IvG=D&YIW.D&OHOH_:V)Qs^m*Nj\*QHdv&btm*N^JusLD&Y\*f%oξnkHNPOHIzRTN^M5m=bQ4380 4560ÁaCE -¤HNPOHQHR QH]PYIRTIN:stO=DkHNPOHQHN-WNPGHILKJOHI]^JQFHGHQFHIRTI&iQJN^IoGHNPRT_1¢ nFHI]PYI\*lYb"WNj\*QHkHQHO=Dm*I]^J&DJIkHPOHIzWNj\*QHYD=RTNPNPRnp(1 − p) = 3600nnÔ!'!"Iom [7.np = 22500 · 0.2 = 4500 np(1 − p) = 22500 · 0.2 · 0.8 = 36004380 − 4500ξn − 45004560 − 4500P = P(4380 ≤ ξn ≤ 4560) = P≤≤606060!=11 Z −x2ξn − 4500≤1 ≈ √e 2 dx = Φ(1) − Φ(−2).= P −2 ≤602π −2Q!¾O=DLkHNPOHQK¯ZbOHYHHQHQWgJIkHYDVQG=D&WOH_Å]PII&JWN^J]^J.WNPOHOHI−2|^JQ stO=DLkHNPOHΦ(x)QKRTILM9OHIyO=D&dJQWy]^J&D1JQH]PJ.QHkHNP]tYHQV JD&v\*QH=DV= j¤c0.0228JWN^JHh0.8413P ≈ 0.8185!b cWde(fhgi]ÍM5~_7 77£ @¾ x%ÄSxTx¢ÔIYDsLDJlHnkJI+W_-FHI\*OHNPOHQHN"G=D&WNPOH]^JW.D¢ W\*NPkHN^J)OHN^sLDoWQH]PQHRTI]^Jl9]PIv_JQHd δ1Wz]^IWIYbFHOHI]^JQA1 , .
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