设随机变量X1,…,Xn,Xn+1独立同分布,且P(X1=1)=p,P(X1=0)=1-p,记: Yi= (i=1,2,…,n). 求.

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问题 设随机变量X1,…,Xn,Xn+1独立同分布,且P(X1=1)=p,P(X1=0)=1-p,记:
    Yi    (i=1,2,…,n).
    求

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答案EYi=P(Xi+Xi+1)=P(Xi=0,Xi+1=1)+P(Xi=1,Xi+1=0)=2p(1-p),i=1,…,n ∴[*]=2np(1-p), 而E(Yi2)=P(Xi+Xi-1=1)=2p(1-p), ∴DYi=E(Yi2)-(EYi)2 =2p(1-p)[1-2p(1-p)],i=1,2,…,n. 若l-k≥2,则Yk与Yl独立,这时cov(Yk,Yl)=0,而 E(YkYk+1)=P(Yk=1,Yk+1=1) =P(Xk+Xk+1=1,Xk-1+Xk-2=1)=P(Xk=0,Xk+1=1,Xk+2=0)+P(Xk=1,Xk-1=0,Xk+2=1) =(1-p)2p+p2(1-p)=p(1-p), ∴cov(Yk,Yk+1)=E(Yk,Yk+1)=EYk.EYk-1 =p(1-p)-4p2(1-p)2, 故[*]=2np(1-p)[1-2p(1-p)]+2[*]cov(Yk,Yk+1) =2np(1-p)[1-2p(n-1)]+2(n-1)[p(1-p)-4p2(1-p)2] =2p(1-p)[2n-6np(1-p)+4p(1-p)-1].

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