Cov(U,V)

设随机变量X，Y相互独立，X～N(0,1)，Y～N(0,4)，U=X+Y，V=X-Y,求：
Cov(U,V)
【正确答案】：

COV(U,V)=E[(U-E(U)(V-E(V))]

&NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP;=E(UV)-E(U)E(V)

&NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP;=E(UV)-0

&NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP;=E(X²)-E(Y²)

&NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP;=D(X)+[E(X)]²-(D(Y)+[E(Y)]²)

&NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP;=1-4

&NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP; &NBSP;=-3

【题目解析】：协方差的计算公式：Cov(X,Y)=E(XY)-E(X)E(Y)据此可得：Cov(U,V)=E[(U-E(U)(V-E(V))] =E(UV)-E(U)E(V) =E(UV)-0 =E(X²)-E(Y²) =D(X)+[E(X)]²-(D(Y)+[E(Y)]²) =1-4 =-3

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