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1SKY:ch71.fYn;n1gÕáÓ©Ù, P(Yn=1)=p0;P(Yn= 1)=q0;p+q=1;p q0;Y0=0;-X0=0;Xn=nXk=1Yk;Un=Xn n(p q);Vn=qpXn;Wn=U2n n[1 (p q)2]:(1)yfUn;n0g;fVn;n0g'ufYn;n0g´;(2)yfXn;n0g'ufYn;n0g´e;(3)¦E(U3jX2)©ÙÆ¿y²E(Un+kjXn)=Un;8n0;(4)¦E(V8jX7=2):)µ(1)ÏEjUnj=EjXn n(p q)j=E nXk=1Yk n(p q) 6nXk=1EjYkj+njp qj=n(1+jp qj)1E(Un+1jY0;Y1;;Yn)=E(Xn+1 (n+1)(p q)jY0;Y1;;Yn)=E(Un+Yn+1 (p q)jY0;Y1;;Yn)=Un+E(Yn+1) (p q)=Un¤±,fUn;n0g'ufYn;n0g´.Ón,éufVn;n0g,ÏEjVnj=EqpXn=nYk=1EqpYn=nYk=1hqpp+qp 1qi=11E(Vn+1jY0;Y1;;Yn)=EVnqpYn+1jY0;Y1;;Yn=VnEqpYn+1jY0;Y1;;Yn=VnEqpYn+1=Vn1=Vn¤±,fVn;n0g'ufYn;n0g´.(3)E(U3jX2)=E(X2+Y3 3(p q)jX2)=X2 3(p q)+E(Y3)=X2 2(p q)2½öµÏfUn;n0g'ufYn;n0g´,¤±µE(U3jX2)=E(U3jY1;Y2)=U2=X2 2(p q):¤±,E(U3jX2)©ÙÆ,E(U3jX2= 2)= 2 2(p q);E(U3jX2=0)=2(p q);E(U3jX2=2)=2 2(p q);P(E(U3jX2)=E(U3jX2= 2))=q2;P(E(U3jX2)=E(U3jX2=0))=2pq;P(E(U3jX2)=E(U3jX2=2))=p2:E(Un+kjXn)=E(Xn+n+kXi=n+1Yi (n+k)(p q)jXn)=Xn+E(n+kXi=n+1Yi) (n+k)(p q)=Xn n(p q)=Un;8n0:(3)E(V8jX7=2)=E((qp)X7(qp)Y8jX7=2)=(qp)2E(qp)Y8=(qp)22.ÅCþSfXn;n0g;÷vEjXnj1; E[Xn+1jX0;X1;;Xn]= Xn+ Xn 1;n0; ; 0:-Yn=Xn+bXn 1;n1;Y0=X0:ÀJÜ· ; ;b¦fYn;n0g'ufXn;n0g´.)µEjYnj=EjXn+bXn 1j6EjXnj+jbjEjXn 1j1:E(Yn+1jX0;X1;;Xn)=E(Xn+1+bXnjX0;X1;;Xn)=E(Xn+1jX0;X1;;Xn)+bXn= Xn+ Xn 1+bXn=( +b)Xn+ Xn 1¦fYn;n0g'ufXn;n0g´,7LkE(Yn+1jX0;X1;;Xn)=Yn;=( +b)Xn+ Xn 1=Xn+bXn 1= +b=1; =b;½b= =1 :33.fYn;n0g;´êÅó,äk=£VÇÝP=(pij);f´Pk.mKS(NÚ¼ê),=f(i)0; f(i)=Xj2Spijf(j);jf(j)jM;i2S:-Xn=f(Yn);KfXn;n0g'ufYn;n0g´.y²µEjXnj=Ejf(Yn)j1;ÏE(Xn+1jY0;Y1;;Yn)=E(f(Yn+1)jY0;Y1;;Yn)=E(f(Yn+1)jYn)=Xj2Sf(j)P(Yn+1=jjYn)=Xj2Sf(j)pYnj=f(Yn)=Xn:ÏdfXn;n0g'ufYn;n0g´.4.Y0=0;Y1;;YnÕáÓ©ÙòzÅCþS,-S0=0;Sn=nPk=1Yk;T=minfn:Sn6 a;½Snbg;T´Ê,Ù¥a0;b0´~ê,KE(ST)=E(Y1)E(T):y²µdK¿½Â ()=E(exp(Y1));qd~??(ØíÑXn= n()exp(Sn)´'ufYn;n1g.e¡ò|^ʽníØ??(Ø,Äky^÷víØ??.E(jXn+1 XnjjY0;Y1;;Yn)=XnEj n()exp(Yn+1) 1j:Ï n()61;9é?¿nT;exp(Sn)6eb:¤±E(jXn+1 XnjjY0;Y1;;Yn)62eb1:y^,e¡yE(T)1:-c=a+b:ÏYn´òz,Ïd,3êN0Ú0;¦PfjSNjcg:V1=SN;V2=S2N SN;;Vk=SkN S(k 1)N:PfTkNg6PfjV1jcgPfjV2jcgPfjVkjcg6(1 )k¤±E(T)=1Xn=1PfTng6N1Xk=0PfTkNg6N=1:4|^ʽníØ??,kE(XT)=E(X1)=1;=E( T()exp(ST))=E( 1()exp(S1))=1:üàé¦0= 0()E(( T T 1())exp(ST))+E( T()STexp(ST))-=0;nE(ST)=E(Y1)E(T)5.bU,«ªqM1,¡VÇ´p,¯,mg ,Ñy¡'Çp uVÇ?):-SnL«cngÑy¡gê,±=AêƯKPfjSn=n pj;nmg:Xi=1L«1igÑy¡,Xi=0;Zn=Sn np=nXk=1(Xk p):|^½Â±y²:Zn´þ0, ÷v p6Zn Zn 61 p:|^í2AzumaØ ª,PfjZnjn;nmg62exp( 2m2);=PfjSn=n pj;nmg62exp( 2m2):6.-f¥m©kxçü¥,zgl¥Ñ¥,£2?ÓôÚ¥.XnL«1ng£ -¥ç¥'~,y²:(1)fXn;n1g´;(2)¦-¥ç¥'~3/5VÇõ?y²µ(1)-Yn=1L«1ngÑç¥,Yn=0L«1ngÑx¥,1ng¥ -¥ç¥Úx¥©OanÚbn,KXn=anan+bn;du0Xn61;ÏdEjXnj1;1n+1gÑx¥VÇbnan+bn;d,an+1=an;bn+1=bn+1;¤±E(Xn+1jY0;Y1;;Yn)=an+1an+1+bnanan+bn+anan+bn+1bnan+bn=an+1an+bn+1anan+bn+anan+bn+1bnan+bn=anan+bn=Xn:5Ïd,fXn;n1g´.(2)d5,E(Xn)=E(X1)=1=2;Ïd,dêÅØ ªPXn356E(Xn)3=5=5=6:7.fXn;n0g´e,y²E(XnjX0;X1;;Xk)Xk;06k6n:y²µ^êÆ8B{.n=k+1,w,¤á.bn=k+m(m1),E(XnjX0;X1;;Xk)Xk:n=k+m+1,E(XnjX0;X1;;Xk)=E[E(Xk+m+1jX0;X1;;Xk+m)jX0;X1;;Xk]E(Xk+mjX0;X1;;Xk)Xk,8n0;E(XnjX0;X1;;Xk)Xk;06k6n:8.fXn;n0g´þ,y² fXn;n0g´e.y²µÏfXn;n0g´þ,d½ÂE(Xn+1jX0;X1;;Xn)6Xn:¤±E( Xn+1jX0;X1;;Xn)= E(Xn+1jX0;X1;;Xn Xn:¤±, fXn;n0g´e.9.fXn;n0g´,y²E(XnjX0;X1;;Xk)=Xk;06k6n:lE(Xn)=C(CØ6n)y²µ^êÆ8B{.n=k+1,w,¤á.bn=k+m(m1),E(XnjX0;X1;;Xk)Xk:n=k+m+1,E(XnjX0;X1;;Xk)=E[E(Xk+m+1jX0;X1;;Xk+m)jX0;X1;;Xk]=E(Xk+mjX0;X1;;Xk)=Xk,8n0;E(XnjX0;X1;;Xk)=Xk;06k6n:íE[E(XnjX0;X1;;Xk)]=E(Xk)6=E(Xn)=E(Xk)=E(Xn)=C;(CØ6n):10.fXn;n0g´pÕáÓ©ÙÅCþS,EjX0j1;T´'ufXn;n0gÊ,EjTj1:y²EhTXk=1Xki=ETE(X0)y²µ-Zn=nPk=1(Xk E(X0));KfZn;n1g´.duE(jZn+1 ZnjjZ1;Z2;;Zn)=E(jXn+1 E(X0)jjZ1;Z2;;Zn)=EjXn+1 E(X0)j6EjXn+1j+EjX0j1Ïd,E(ZT)=E(Z1)=0Ó,E(ZT)=ETXk=1(Xk E(X0))=ETXk=1Xk ETE(X0)=EhTXk=1Xki ETE(X0)¤±,EhTXk=1Xki=ETE(X0)11.Yn=a+X1+X2++Xn;fXigÕáÓ©ÙÅ, E(Xi)=0:¡Yn´Åþ}.÷vE(X2i)=2+1, Zn=Y2n var(Yn)=Y2n n2;@o,fZn;n1g'ufXng´.y²µdK¿,E(Xn)=0;var(Yn)=n2E(Xn+1jX0;X1;;Xn)=E(Xn+1)=0E(X2n+1jX0;X1;;Xn)=E(Xn+1)2=27u´E(Y2n+1jX0;X1;;Xn) Y2n=E((Yn+Xn+1)2jX0;X1;;Xn) Y2n=[Y2n+2E((YnXn+1)jX0;X1;;Xn)+2] Y2n=2YnE(Xn+1jX0;X1;;Xn)+2=2:ddE(Zn+1jX0;X1;;Xn)=E(Y2n+1jX0;X1;;Xn) (k+1)2=Y2n k2=Zn¤±,fZn;n1g'ufXng´.12.fXn;n0g´Ke,Ké?¿0kP(max06k6nXk)6E(Xn);n0y²µé0;-T=(n;é¤k0;1;2;n;Xk6minfk:Xkg;Ù¦KT´Ê.é?½n,Ïn;TþÊ, nT,ÏdE(Xn)E(XT)E(XTIfmax06k nXkg)E(Ifmax06k nXkg)=P(max06k nXk)Ïd,é?¿n0;P(max06k6nXk)6E(Xn):13.fXn;n0g´,Ké?¿0kP(max06k6njXkj)6EjXnj;n0y²µdufXn;n0g´,Kén0,kE(jXn+1jjX0;X1;;Xn)jE(Xn+1jX0;X1;;Xn)j=jXnjÏd,fjXnj;n0g´Ke,|^þK¥'uKe(Ø,P(max06k6njXkj)6EjXnj;n0:814.fXn;n0g´ÕáÓ©ÙÅS,é?¿n0;E(Xn)=0;E(X2n)=21;-Y0=0;Yn=nXk=1Xk;n1:K,é?¿0;kP(max06k6njYnj)6n22:y²µd½ÂfYn;n0g´,´yfY2n;n0g´e,Ïd,dKe(Ø,é?¿0;kP(max06k