面試數學基礎100問

概率論與數理統計

X、Y爲兩個獨立的隨機變量,其各自的期望,方差均已知,D(XY)=?

D(XY) = E{[XY-E(XY)]^2}

= E{X²Y²-2XYE(XY)+E²(XY)}

= E(X²)E(Y²)-2E²(X)E²(Y)+E²(X)E²(Y)

= E(X²)E(Y²)-E²(X)E²(Y)

如果 E(X) = E(Y) = 0,

那麼 D(XY) = E(X²)E(Y²) = D(X)D(Y)，

也就是說當 X,Y獨立，且X,Y的數學期望均爲零時，X,Y乘積 XY的方差D(XY)等於：

D(XY) = D(X)D(Y)