概率论教程部分答案(钟开莱)

1.11.flimx!¡1f(x)=1Pn=1bnlimx!¡1±an(x)=1Pn=1bn¢0=0limx!1f(x)=1Pn=1bnlimx!1±an(x)=1Pn=1bn¢1=1Pn=1bn2.f(x)=[x]1Pn=1±n(x)¸0;8x2Rfx0f(x)=1Pn=1±n(x)¡1Pn=1±n(¡x)¡±0(¡x)3.nB¡Afajgnj=1a1¢¢¢an8xan;ya1f(x)¡f(y)=f(x)¡f(an)+nXj=1[f(aj)¡f(aj¡)]+n¡1Xj=1[f(aj+1¡)¡f(aj)]+f(a1¡)¡f(y)¸nXj=1[f(aj)¡f(aj¡)]¢nB¡A1f8n¸11nfA=fxjf(x+)¡f(x¡)0g1nAn=fxjf(x+)¡f(x¡)1ngA=1[n=1Anffn=8:f;jxj·nf(n);xnf(¡n);x¡nffn1.21.xFlim#0[F(x+)¡F(x¡)]=F(x+)¡F(x¡)=0x2.800·Xx¡ajx[F(aj)¡F(aj¡)]·F(x¡)¡F(x¡)#0Px¡aj·x[F(aj)¡F(aj¡)]=Px¡ajx[F(aj)¡F(aj¡)]+[F(x)¡F(x¡)]Px¡aj·x[F(aj)¡F(aj¡)]x[F(x)¡F(x¡)]2fc8x2R;80Fc(x)¡Fc(x¡)=[F(x)¡F(x¡)]¡[Fd(x)¡Fd(x¡)]=[F(x)¡F(x¡)]¡[Xjbj±aj(x)¡Xjbj±aj(x¡)]=[F(x)¡F(x¡)]¡Xx¡aj·xbj#0fc31.26.xF(x+)¡F(x¡)080F(x+)¸F(x+);F(x¡)·F(x¡)F(x+)¡F(x¡)¸F(x+)¡F(x¡)0xx±08y2^U(x;±);9y0;F(y+y)=F(y¡y)yF(x;x+±)F(x+)(x¡±;x)F(x¡)xF(x+)F(x¡)xfbng1n=101Pn=1bn=1F(x)=1Pn=1bn±an(x)fang1n=11.31.F0F=Fs,Fac(x)=Zx¡1F0(t)dt=0;8x2R,F0(x)=0;a:e:,Fsingular1F=Fac,F(x)=Zx¡1F0(t)dt;8x2R,F(x0)¡F(x)=Zx0xF0(t)dt;8xx02R,Fabsolutelycontinuous2.1.2.3FF1F2F21.3.1cF2=Fac+Fs¯=Fac(1)Fc¯F3+(1¡¯)F4F3=1¯Fac;F4=11¡¯FsF=®F1+(1¡®)¯F3+(1¡®)(1¡¯)F4F26.F(x)=1Pn=1bn±an(x);fbng1n=10;1Pn=1bn=1F0=0;a:e:8x=2fang1n=1±0aj(x)=0;8jFF0(x)=1Pn=1bn±0an(x)=02.13.TkCknk=1;¢¢¢;nTnPk=0Ckn=22nA1;¢¢¢;An1.Ajn¡1Ai;i6=j2.fAjgBj=(\i2IjAi)\(\i2NnIjAci)N=f1;¢¢¢;ng;IjN2nBj¾(fAjg)=¾(fBjg)96.­=NF1=¾(f1g);Fj+1=¾(Fj;fj+1g);j¸1;F=1[j=1FjFj½Fj+1An=f2ng2F2n½F;n¸1A=1[n=1An=2FF¾1_j=1Fj=¾(1[j=1Fj)Fj¾9.G=fAjA=[n2K¤n;KgG½F8A=[n2K¤n2GAc=\n2K¤cn=\n2K[i6=n¤i=[n2Kc¤n2GGAj=[n2Kj¤n2G;j¸11[j=1Aj2GGG¾¤n2G;8n¸1F½G32.26.E=¢TF1=¢TF2;F1;F22F(F1SFc2)T¢=(F1T¢)S(Fc2T¢)=(F2T¢)S(Fc2T¢)=¢F1SFc2¾¢F1SFc22FP(F1SFc2)=1·P(F1)+P(Fc2)P(F1)¸P(F2)P(F1)·P(F2)P¤P¤P¤(¢)=P(­)=1fEn=¢TFngn¸1Fn2FF01=F1;F0n=FnTFcn¡1¢¢¢TFc1;n1,E0n=¢TF0n;n¸1P¤(PnEn)=P¤(¢TSnFn)=P¤(¢TPnF0n)=P(PnF0n)=PnP(F0n)¢TFiTFj=;;8i;j¸1!¢½(FiTFj)c!P(FiTFj)=0PnP¤(En)=PnP(Fn)=PnP(FnTFcn¡1T¢¢¢TFc1)+P(FnT(Fcn¡1T¢¢¢TFc1)c)=PnP(F0n)+0.P¤(PnEn)=PnP¤(En)9.C¾FfEjg(1[j=1Ej)c,E0Ejfpjg¸0;1Pj=0pj=11P(Ej)=pj;j¸1;P(E0)=p0PFT:F!2­T(Ej)=f!j+1g;j¸0T([j2IEj)=[j2If!j+1gT11.¹f(x¡1n;x]gn¸1¹((x¡1n;x])=F(x)¡F(x¡1n)n!1¹(x)=F(x)¡F(x¡)x¹F¹¾fEngR¹En¹n(A)=¹(ATEn),limn!1¹n(A)=¹(A)¹n(x)=Fn(x)¡Fn(x¡)¹(x)=F(x)¡F(x¡)x¹¹n12.²F(x)=Pibi±ai(x)faigA2B(R)ai1¸¹(ASfaigi¸1)=¹(A)+¹(faigi¸1)=¹(A)+1¹(A)=0¹2¹faigi¸1F(x)=¹((¡1;x])=¹((¡1;x]\faigi¸1)+¹((¡1;x]\faigci¸1)=¹(faijai·xg)+0=Xi¹(ai)±ai(x)F²¹,¹(fxg)=F(x)¡F(x¡)=0;8x2R,F22.²x11:P(x)0E2FfxgE=fxgSNN18F½E;F2Fx=2NP(F)=P(FTE)=P(FTfxg)+P(FTN)=P(FTfxg)=0P(E)x22:²E22:f(x)=P(ET(¡1;x])f(¡1)=0;f(1)=P(E)0fP(E)0x0f(x0)=12(P(E)+0)x0f(x)¡f(x¡)=P(fxgTE)!x2EP(fxg)=P(E)fxgEx11:24.¹S8fxng½S;n¸1;xn!x0x03B(x0;±)nxn2B(x0;±)n±00B(xn;±0)½B(x0;±)¹(B(x0;±))¸¹(B(xn;±0))0x02SSc8x2Sc±x0B(x;±x)½Sc¹(B(x;±x))=0Sc=[x2ScB(x;±x)LindelofSc=[nB(xn;±xn)¹(Sc)·Pn¹(B(xn;±xn))=0O¹(O)=08x2O;B(x;±)½O¹(B(x;±))·¹(O)=0x2ScO½Sc¹B180;¹(B(x;))¸F(x+2)¡F(x¡2)F(x+)¡F(x¡)¸¹(B(x;))43.14.G(µ(!))xyxF(y)·µ(!)F(x)·µ(!)f!jG(µ(!))xg½f!jF(x)·µ(!)gµ(!)F(x)0µ(!)F(x+)G(µ(!))¸x+xf!jµ(!)F(x)g½f!jG(µ(!))xgf!jµ(!)F(x)g½f!jG(µ(!))·xg½f!jµ(!)·F(x)gP(G(µ)·x)=P(µ·F(x))=F(x)11.FfXgX¾fX¡1(B)jB2B1gXBX=c¤=­Ac3.21.X1¤¸0;a:s:RX1¤dP=0X1¤=0;a:s:¤X=0;a:s:12.80P(jX1¤nj)=RjX1¤njdP=RfjXjg\¤ndP·P(¤n)!0;n!1X1¤nP!0Xlimn!1R¤nXdP=R0dP=0P(jXjn)·EjXjn!0¤n=fjXjng3.º(­)=1;º(?)=0fAng1n=1½F1Pn=1º(An)=1Pn=11ARX1AndP=1ARX1Pn=11AndP=1ARX11Sn=1AndP=º(1Sn=1An)6.fY¡XngFatouR¤limn!1(Y¡Xn)dP·limn!1R¤(Y¡Xn)dPYR¤YdPYFatouR+Xn=n1[0;1n);n¸1limn!1Xn=0;a:s:RXndP=1;8n¸1supnXn=n1[1n+1;1n)EsupnXn=Pnn(1n¡1n+1)=123.211.EjXj1fjXj¸¸ag·(EX2)12(E1fjXj¸¸ag)12=P(jXj¸¸a)12EjXj1fjXj¸¸ag=EjXj¡EjXj1fjXj¸ag¸a¡¸a=(1¡¸)aP(jXj¸¸a)¸(1¡¸)2a212.Cr8a;b2R;p0;ja+bjp·maxf1;2p¡1g(jajp+jbjp)16.FubiniZ1¡1(F(x+a)¡F(x))dx=Z1¡1P(xX·x+a)dx=Z1¡1Z­1fxX·x+agdPdx=Z­Z1¡11fxX·x+agdxdP=Z­adP=a1Lebesgue¡Stieltjes17.F(0¡)=0FubiniZ10(1¡F(x))dx=Z1¡1P(Xx)dx=Z1¡1Z­1fxX(!)gdP(!)dx=Z­Z1¡11fxX(!)gdxdP(!)=Z­XdP=Z10xdF(x)Xr:v:FLebesgue0EX=R1¡1P(Xx)dx=R1¡1P(X¸x)dx3.32.P(X1X2=§1)=12X1;X2;X1X2P(X1=1;X2=¡1;X1X2=¡1)=146=18=P(X1=1)P(X2=¡1)P(X1X2=¡1)X1;¢¢¢;Xn¡1Xn=X1X2¢¢¢Xn¡1n¡1P(X1=¡1;Xi=1;i=2;¢¢¢;n)=06=P(X1=2¡1)n¦i=2P(Xi=1)n=3;¤1=fX1=1g;¤2=fX2=1g;¤3=fX3=¡1g¤2S¤3=¤1S¤2P(¤1T(¤2S¤3))=P(¤1)6=P(¤1)P(¤2S¤3)3.Afi1;¢¢¢;ingFijEc®kP(\jFij)=¦jP(Fij)k=0E®k¡1k(k·n)Fij=Ecj;j=1;¢¢¢;kP(\jFij)=P(\j·k¡1Ecj\Eck\\jkEj)=P(\j·k¡1Ecj\\jkEj)¡P(\j·k¡1Ecj\Ek\\jkEj)=¦j·k¡1P(Ecj)¦jkP(Ej)¡¦j·k¡1P(Ecj)P(Ek)¦jkP(Ej)=¦j·kP(Ecj)¦jkP(Ej)=¦jP(Fij)fF®gBf1;¢¢¢;ngP(n\i=1([®2AiE®)c)=P(n\i=1\®2AiEc®)=n¦i=1¦®2AiP(Ec®)=n¦i=1P(\®2AiEc®)=3n¦i=1P(([®2AiE®)c)43.35.®2AAA¯;A=[¯2¦A¯f¾(X®;®2A¯);¯2¦g¦f1;¢¢¢;ngC=fB2¾(X®;®2A1)jP(BT\2·k·nBk)=P(B)¦2·k·nP(Bk);8Bk2X¡1®(B);®2Akg;C0=f\iCijCi2¾(Xi);i2A1gC¸C0¼C¾C0C¾¾(C0)¾(C0)=¾(X®;®2A1)C=¾(X®;®2A1)¾(X®;®2A1)X¡1®(B);2·k·n;®2Ak2·k·n¾(X®;®2Ak);1·k·n10.X;YE(jX+Yjp)=RRjx+yjp¹2(dx;dy)=RRjx+yjp¹x(dx)¹y(dy)Rjx+yjp¹x(dx)1;¹y¡a:e:y0Rjx+y0jp¹x(dx)1CrE(jXjp)=Rjxjp¹x(dx)·maxf1;2p¡1g(Rjx+y0jp¹x(dx)+Rjy0jp¹x(dx))=maxf1;2p¡1g(Rjx+y0jp¹x(dx)+jy0jp)114.8x2(0;1];1¡pp1nxnP(²n=1)=p;P(²n=0)=1¡p18.[0;1]A=2¹BfagA£fagA£fag=2¹B£¹BA£fag½[0;1]£fagR2LebesgueA£fag2B£B4.11.8M0;P(XnM;i:o:)=0,8m0;P(\n[k¸nfXkm