设X,Y为两个随机变量,E(X)=E(Y)=1,D(X)=9,D(Y)=1,且ρXY=,则E(X一2Y+3)2=__________.

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问题 设X,Y为两个随机变量,E(X)=E(Y)=1,D(X)=9,D(Y)=1,且ρXY=,则E(X一2Y+3)2=__________.

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答案25

解析 E(X一2Y+3)=E(X)一2E(Y)+3=2,
D(X一2Y+3)=D(X一2Y)=D(X)+4D(Y)一4Cov(X,Y),
由Cov(X,Y)=×3×1=一2,得
D(X一2Y+3)=D(X)+4D(Y)一4Cov(X,Y)=9+4+8—21,
于是E[(X一2Y+3)]2=D(X一2Y+3)+[E(X一2Y+3)]2=21+4=25.
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