x,y相互独立时,方差d(xy)

Question

x,y相互独立时,方差d(xy)

Answer 1

D(XY) = E{[XY-E(XY)]^2} (1)
= 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) (2)
如果 E(X) = E(Y) = 0,
那么 D(XY) = E(X²)E(Y²) = D(X)D(Y)， (3)
也就是说当 X,Y独立，且X,Y的数学期望均为零时，X,Y乘积 XY的方差D(XY)等于：
D(XY) = D(X)D(Y). (4) //: 就是(3)式