А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 14
Текст из файла (страница 14)
. .]PII&JWN^J]^J.WNPOHO=IHn*]PFHG=D&WNjmH\*QHW.D5ZEIGHR¦bL\Dp0 , p 1 , . . .ξψξ (z) = Ez =∞Xz n pn ,n=0QsYI&JIGHId"]j\*Njm=bN^JnkJIFHGHIQstWItm=KiDK"ZbOHYHHQK"]j\=bk=D&dHOHId'WN\*QHkHQHOH_NP]^JlOHNkJIEQHOHI&N&nYD&YFHGHIQstWILm=KiDKZbOHYHHQKFHI]j\*Njm*IW.DJNj\*lOHI]^JQzk=D&]^JOH_:VstO=DokHNPOHQHdXWNPGHILKJOHI]^JQLRTNPOHOHI%FHI%|^JIdXFHGHQHkHQHOHNaFHGHIQstWItm=KiDKEZbOHYop0 , p 1 , .
. .HQKuIFHGHNjm*Nj\*NPO=D\*QplmH\=KuHNj\*IkHQH]j\*NPOHOH_:V]j\=bk=D&dHOH_:VWNj\*QHkHQHOGHIQstWItm=KiDKSZbOHYHHQKIv\DmD&N^Jc]j\*Njm=bf[iQ=RTQI]POHIWOH_-RTQS]PWIdH]^JW.D&RTQh ψξ (eit ) = ϕξ (t)~[ "NPG=]jD&\*WQ NPOHξ]^JQ WI η ]^bJlOHN^sLD&WQH]PQHRT_-Ny]j\=bk=D&dHOH_-NyWNj\*QHkHQHOH_nJIq]PFHG=D&WNjmH\*QHWIψξ+η (t) = ψξ (t)ψη (t);@ "NP]j\*QnE|ξ|k < ∞ k ≥ 1nHJI(k)ψξ (z)= Eξ [k] .@U*U';*)!^"?*¾^0EH!E=!<'U^hCBS'*!"EH?+HhC*+;<'7 .[*)-;+!><*A)+BU^mA)+\!'BHE><1!*1Q\'!><'!!$5R ¤Á ux¯ EξkξϕξFξψξPξfξÏd»¼±º7ì¾ £z=1 JD&Yn*RT_1FHI&stO=D&YIRTQ\*QH]Pl9]PI9W]PNPRTQI]POHIWOH_:oRTQuVD&G=D&YJNPGHQH]^J.QHYD&RTQq]j\=bk=D&dHOH_:V)WNj\*QHkHQHO)Q+QVG=D&]PFHGHNjm*Nj\*NPO=QHdCEDgGHQH]^bOHYN FHIYDsLD&O=D§m*Q=DoUPG=D&RTRaD-QHRTF\*QHYD&HQHdmH\=KEIFHGHNjm*Nj\*NPOHOH_VXO=D&RTQVDoG=D&YJNPGHQH]^J.QHY]j\=bk=D&dHOH_:V¸WNj\*QHkHQHO:çJGHNj\*YHQFHIoYDst_-W.D&f[JLn9YD&YHQHNVD&G=D&YJNPGHQH]^J.QHYHQ IFHGHNjm*Nj\=KHf[Jm*GbUPQHNSItm*OHI&stO=DLkHOHIHn=NP]j\*Q]^biNP]^J.Wbf[J»]j\*Njm=bf[iNPdgJD&v\*QHHNyFHGHQHWNjm*NPOH_ÂZEIGHR¦bt\*_mH\=KEVD&G=D&YJNPGHQH]^J.QHkHNt]PYHQVQXFHGHIQstWILm=KiQVXZbOHYoHQHduI]POHIWOH_:VuG=D&]PFHGHNjm*Nj\*NPOHQHds ðt Þ x ÞÛ¥z« x ®Ù-zL®ß«PÙÛH®«® Ø x Ú®=Ý^ªÚe¬¥yS®«VÇÆHÛH®H®Pξϕξ (t)ψξ (z)â ãrâ aξ ár=eitazaBi(n, p)(1 + p(eit − 1))n(1 + p(z − 1))nP ois(λ)eλ(e −1)1−p1 − peiteitb − eitait(b − a)sin atatiteλ(z−1)1−p1 − pzG(p)R[a, b]R[−a, a]N (a, σ 2 )exp(λ)K(a, σ)S{{{σ 2 t2eita− 2λλ − iteita−b|t|éZ§¨¨ÁË§ì¾ £{{¾ x axCE )Ô¢&¡IGHRaD\*lOHIHnFHGHIQstWItm=KiDKSZbOHYHHQKXmH\=KSW_-GHILM5m*NPOoOHIdy]j\=bk=D&dHOHIdWNj\*QHkHQHOH_1OHNEIFHGHNjm*N\*NPO=Dn*NP]j\*QkHQH]j\*I OHNKHW\=KHN^J]rKHNj\*_-R-a¾x%ÄSxax ÄSIYDsLDJlHn¦kJI)ZbOHYHHQKKHW\=KHN^J]rK`VD&G=D&YJNPGHQH]^J.QHkHNjot]PYIduZbOHYHHQHNPdOHNPYI&JIGHId]j\=bk=D&dHOHIdWNj\*QHkHcosQHOH_ÁaCE -I]^JGHIQHRG=D&]PFHGHNjm*Nj\*NPOHQHN]j\=bk=D&dHOHIdqW&Nj\*QHkHQHOH_ÇJD&YIN&nkJINPUPIVD&G=D&YJNPGHQH]PJQ=kHNP]PYId1ZbOHYHHQHNPdêKHW\=KHN^J]rK%s ]PII&JOHI&pNPOHQK; cos tW_JNPYD&N^JnkJIom [cos t =eit + e−it11= · e1·it + · e−1·it .222¾ D&RTN^JQHR-n:kJI`FHG=D&W.DK¸k=D&]^JlFHI]j\*Njm*OHNPUPIG=D&WNPOH]^JW.D`NP]^Jl§OHNkJIQHOHIN&nYD&YVD&G=D&YJNPGHQH]^J.QHkHNt]PYDK)ZbOHYHHQK)]PQHRTRTN^JGHQHkHO=IUPIyv&QHOHIRTQ=D\*lOHIUPI"G=D&]PFHGHNjm*Nj\*NjoOHQK¾x%ÄSxax» =ÄSIYDsLDJlHnkJIyZbOHYHHQK −i|t| OHNRTILMN^J'v_Jl9VD&G=D&YJNPGHQo]^JQHkHNP]PYIdZbOHYHHQHNPd]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ eÁaCE -aÄE\=Km*IYDsLDJNj\*l]^JW.D bJWNPG.M5m*NPOHQK¸sLDmDLkHQ¨WI]PFHI\*l&sPbNPRT]K]PWIdH]^JWIR»+VD&G=D&YJNPGHQH]^J.QHk=NP]PYId¨ZbOHYHHQHQGHNjm*FHI\*IM9QHR-nTkJI −i|t| NP]^JlVD&G=D&YJNPGHQH]^J.QHkHNt]PYDKZbOHYHHQKuOHNPYI&JIGHId]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ =eIUemDom [ξϕ−ξ (t) = ϕξ (−t) = e−i|t| ,∀ t ∈ IR,b!cWdef0g7iFj ¢.JIXNP]^JlXG=D&]^FHGHNjm*Nj\*NtOHQK Q]PIWF=DmD&f[J"FHGHQ5|^JIR UPIWIGKJnkJIE]j\=bk=D&dHO=DKξ −ξWNj\*QHkHQHO=D QHRTNPN^Jr j.CI[JIUemD[mH\=KYIRTF\*NPY]POHI]PIFHGKMNPOHOHξId'VD&G=D&YJNPGHQH]^J.QHkHNt]PYId+ZbOHYHHQHQQHRTNPN^J9RTNP]^JIzG=D&WNPOH]^JWI'< @ '!><hA)+B!'ϕξ (t) = ϕξ (t).ED&YqQstWNP]^JOHIHnYIRTF\*NPY]POHI]PIFHG.KMNPOHOHIN5kHQH]j\*IuG=D&WOHIy]tD&RTIR¦b)kHQH]j\=b+JIUrmDQ)JI\*lYI'JIUemDnYIUemD'YIRTF\*NPY]POHIN5kHQH]j\*IyKHW\=KHN^J]rK)m*NPdH]^JWQJ.Nj\*lOH_%RÀ|^JIHnW5k=D&]^JOHI]^JQn*]j\*Njm=bN^J"Qs9 e j*ça\*Njm*I&W.DJNj\*lOHIHn*RT_êFHGHQVItm*QHR¨YyFHGHI&JQHWIGHNjokHQHfSn=FHI]PYI\*lYb'ZbOHYHHQK −i|t| OHNKHW\=KHN^J]rKym*NPdH]^JWQJN^\*lOHIde¾x%ÄSxTx} @ GHIQstWItm=KiDKZbOHYHHQKOHNPYI&JIGHId`HNj\*IkHQH]j\*NPOHO=Id]j\=b.ok=D&dHOHIdEWNj\*QHkHQHOH_ nPFHGHQHOHQHRDf[iNPdS]TFHI\*ILM9QJNj\*lOH_-RTQXWNPGHIKJOHI]jJKHRTQ\*QplOHNPI&JGHQH=DJ.Nj\*lOH_-NXξstO=DLkHNPOHQKn*G=D&WO=D=CED&dJQQψ(z)Eξ DξÁaCE -:I]PFHI\*l&sPbNPRT]K"]PWIdH]^JWIR @ FHGHIQstWItm=KiNPd'ZbOHYHHQHQRTNjoNPR[om Eξ = Eξ [1] = ψ 0 (z)z=1,Dξ = Eξ 2 − (Eξ)2 = Eξ(ξ − 1) + Eξ − (Eξ)2 = Eξ [2] + Eξ(1 − Eξ) == ψ (2) (z)z=1+ ψ 0 (z)z=11 − ψ 0 (z)z=1.¾ x%ÄSxTxÀ £Hb.]^JlQ{ûOHN^sLD&WQH]PQHRT_-Nz]j\=b.oξ ∼ P ois(λ1 ) η ∼ P ois(λ2 )k=D&dHOH_-NXWNj\*QHkHQHOH_CED&dJQuG=D&]PFHGHNjm*Nj\*NPO=QHNX]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ζ =ξ+ηÁaCE -IXJNPIGHNPRTN -m*I]^J&DJIkHOHIO=D&dJQ"WQm9VD&G=D&YJNPGHQH]^J.QHkHNt]PYIdZbOHYHHQHQCED&dm*NPR`VD&G=D&YJNPGHQH]^J.QHkHNt]PYbfZbOHYHHQHf]j\=bk=D&dHOHIdyWNj\*QHkHQoOH_ =I5JϕN^ζI(t)GHNPRTN Ö EQHRTNPNPRom ξϕξ (t) = Eeitξ=∞Xk=0kitk −λ1 λ1e ek!=e−λ1∞X(λ1 eit )kk=0k!it= e−λ1 eλ1 e = eλ1 (eit −1).I5]PWIdH]^JW&b Ö RaDJNPRaDJQHkHNP]PYIUPI9ILM9QmD&OHQKϕζ (t) = ϕξ (t)ϕη (t) = e(λ1 +λ2 )(eit −1).¾ D&RTN^JQHR-nkJI]^bRTRaDOHN^sLD&WQH]PQHRT_:V'Fb.D&]P]PIOHIW]PYHQV']j\=bk=D&dHOH_:V'WNj\*QHkHQHO'QHRTNjoN^JzJD&YMNEFbD&]P]PIOHIW]PYINXG=D&]PFHGHNjm*Nj\*NPOHQHN&s ðt Þ x ÞÛ¥z« x ®Ù-zL®ß«PÙÛH®«® Øx Ú®=Ý^ªÚe¬¥yS®«VÇÆHÛH®H®M5~_7 77 Ö¾x%ÄSxaxÇÔ¢ÄSIYDsLDJl5GHN^sPbL\*lLJ&DJ_En*FHGHQHWNjm*NPOHOH_-NSWJ&D&v\*QHHNS ¾x%ÄSxax ÄSI&YDsLDJlHnHkJI5ZbOHYHHQKOHNRaILMN^J5v_Jl5FHGHIQs^oWItm=KiNPduZbOHYHHQHNPd]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ ψ(z) = |z|¾x%ÄSxaxÔb]^Jl Q {¥]j\=bk=D&dHOH_-NcWNj\*QHkHQHOH_nFHGHQHkHNPR FHGHQHOHQHRaD&N^JstO=DLkHNPOHQK Q ]:WNPGHILKJOHξI]^JKHηRTQYDM5m*IN&nD {1stO=DkHNPOHQK n ξn n ]:WNPGHIoη1 2 3KJOHI]^JLKHRTQ 0 1 nnn]PII&1/2JWN^J]^J.WNPOHOHIHHÄSIYDsLDJlHnHkJIz0OHN]^biNP]^J.W&bNPJ]j\=bk=D&dHOHId1/8WNj\*QHkH1/4QHOH_ 1/2nHOH1/8NsLD&WQH]KiNPduI&J QyJD&YIdn=kJIζξξ+ζ =η¾x%ÄSxax} Ö b.]^JlQ{ÑOHN^sLD&WQH]PQHRT_-N"]j\=bk=D&dHOH_-N"WNj\*QHkHQHOH_nQHRTNjof[iQHN]^J&D&OmD&GJOHIN+OHIGHRaξD\*lOHηING=D&]PFHGHNjm*Nj\*NPOHQHNaÄSIYDsLDJlHnkJI`]j\=bk=D&dHO=DKWNj\*QHkHQHO=D 2QHRTNPN^J9FHIYDsLDJNj\*lOHINXG=D&]PFHGHNjm*Nj\*NPOHQ=NX]EF=D&G=D&RTN^JGHIRξ + η21/2¾x%ÄSxaxæ CED&dJQ¸G=D&]PFHGHNjm*Nj\*NPOHQHKYI&JIGH_-RÀ]PII&JWNjJ]PJW&bfcJ§]j\*Njm=bf%oiQHNVD&G=D&YJNPGHQH]^J.QHkHNP]tYHQHNSZbOHYHHQHQhD. ~Hv cos2 texp{−t2 }¾x%ÄSxax ÄSIYDsLDJlXI]POHIW&OH_-N%]PWIdH]^JW.DVD&G=D&YJNPGHQH]PJQHk=NP]PYId9ZbOHYoHQHQ¾x%ÄSxaxÇ ÄSIYDsLDJlzI]POHIWOH_-NE]PWIdH]^JW.DzFHGHIQsPWILm=KiNPdZbOHYHHQHQ¾x%ÄSxax¥ *b.]^Jl Q { OHN^sLD&WQH]PQHRT_-N]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_nFHGHQokHNPRFHGHQHOHQHRaD&N^JstO=DLkHξNPOHQK η n Q ]WNPGHILKJOHI]^JKHRTQYDM5m*IN&.ÄSIYDosLDJlHξn=k+JηIzILm*O=DQsXWNj\*QHkHQHO Q\*0Q 1 QHRT2NPN^J9W_-GHILM5m*NPOHOHINEG=1/3D&]PFHGHNjm*Nj\*NPOHQ=N&ξη¾x%ÄSxax; @ CED&dJQqG=D&]PFHGHNjm*N\*NtOHQK YI&JIGH_-R¯]PII&JWN^J]^JW&bf[J]j\*Njm=bf%oiQHNEFHGHIQstWILm=KiQHNXZbOHYHHQHQh*D. ~=v (1 + z)2 /4p(1 − (1 − p)z)−1¾x%ÄSxax} t£HÄSIYDsLDJlHnkJIyVD&G=D&YJNPGHQH]^J.QHkHNP]tYDKZbOHYHHQK`kHN^JO=DuJIoUemD5QyJI\*lYI5JIUrmDn=YIUrmD5]PII&JWN^J]^JW&bf[iDKZbOHYHHQKG=D&]PFHGHNjm*Nj\*NPOHQKF (x)btm*IW\*N^JWIGKHN^J9]PII&JOHI&pNPOHQHfF (x) = 1 − F (−x − 0)¾x%ÄSxax ¢&ÄSIYDsLDJlHn.kJIEVD&G=D&YJNPGHQH]^J.QHk=NP]PYDK9ZbOHYHHQK9WN^iNP]^JWNPOoO=DJIUrmD5QyJI\*lYI5JIUemDn*YIUemD5IO=DkHN^JO=D¾x%ÄSxax¸ LtÄSIYDsLDJlHn&kJIkHNjJO=DKVD&G=D&YJNPGHQH]^J.QHkHNt]PYDK5ZbOHYHHQKϕ(t)FHGHNjm*]^JD&WQHRaDzW5WQm*N[ϕ(t) =Uem*N+∞Zcos tx dF (x),−∞{ ]PII&JWN^J]^JW&bf[iDKZbOHYHHQKG=D&]PFHGHNjm*Nj\*NPOHQK}¾x%ÄSxax¿ *b]^Jl Q {ÂOHN^sLD&WQH]PQHRT_-NSItm*QHO=D&YIWIzG=D&]PFHGHNjm*Nj\*NPOHOH_%N]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_¿]XVD&ξG=D&YηJNPGHQH]^J.QHkHNt]PYId ZbOHYHHQHNPdCED&dJQ+VD&G=D&Yoϕ(t)JNPGHQH]^JQ=kHNP]tYbf¥ZbOHYHHQHfÇ]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ξ−η¾x%ÄSxaxÇ Ö Hb.]^Jl{}ZbOHYHHQKuG=D&]PFHGHNjm*Nj\*NPOHQK]VD&G=D&YJNPGHQH]^J.QoF (x)kHNP]PYIdZbOHYHHQHNPdLÄSIYDsLDJlHnkJI KHW\=KHN^J]rKVD&G=D&YJNPGHQH]^J.QHkHNt]PYIdϕ(t)ϕ(t)ZbOHYHHQHNPdn*QuO=D&dJQu]PII&JWN^J]^J.W&bf[ibf¥ZbOHYHHQHf¿G=D&]PFHGHNjm*Nj\*NPOHQKF (x)b!cWdef0g7iFj ¢¾ x%ÄSxTxÇ L=b]^Jl{}ZbOHYHHQKuG=D&]PFHGHNjm*Nj\*NPO=QK]cVDG=D&YJNPGHQH]PJQoF (x)kHNP]PYId'ZbOHYHHQHNPdÄSIYDsLDJlHnkJI2 KHW\=KHN^J]rK"VD&G=D&YJNPGHQH]^J.QHkHNt]PYIdϕ(t)ZbOHYHHQHNPdnQyO=D&dJQ]PII&JWN^J]^JW&bf[iSbf¿|ϕ(t)|ZbOHYHHQHf¿G=D&]PFHGHNjm*Nj\*NPOHQHK¾x%ÄSxTx¥ b]^Jl Q {ÂOHN^sLD&WQH]PQHRT_-NItm*QHO=D&YIWIzG=D&]PFHGHNjm*Nj\*NPOHO=_-N]j\=bk=D&dHOH_-N-WNj\*QHkHQHOH_n&QHRTNPξf[iQHηN]PQHRTRTN^JGHQHkHO=ING=D&]PFHGHNjm*Nj\*NPO=QHN&nFHGHQHkHNPRξ −ηQItm*QHO=D&YIWI5G=D&]PFHGHNjm*Nj\*NPO=_CEDdJQG=D&]PFHGHNjm*Nj\*NPOHQHN [|ξ| − |η|ξø÷ çß#" ßm ãÛä竪 ãã õ èÑÏè竪§«·¼Ï æ¿Ð竪 ãò!æóñBÊõ ø ùûúýüËþ¿ÿ b]^Jl"sLDmD&OHI"OHNPYI&JIGHINWNPGHILKJOHI]^JOHINFHGHI]^JG=D&OH]^JWIQ+]j\=b.o(Ω, F, P)k=D&dHOH_-NEWNj\*QHkHQHOH_O=D5OHNPR-ξ1 , .
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