设离散型随机变量X的数学期望为EX,方差为DX,试证明：DX=EX^2-(EX)^2

题目

设离散型随机变量X的数学期望为EX,方差为DX,试证明：DX=EX^2-(EX)^2

答案

证明：
D(X)=E{[X-E[X]]^2}(方差的定义）
=E{X^2-2*X*E[X]+E[X]^2}
=E[X^2]-E{2*X*E[X]}+E{E[X]^2}
=E[X^2]-2*E[X]*E[X]+E[X]^2
=X[X^2]-E[X]^2