设随机变量X的期望为E(x),方差为D(x)>0,X*=(X-E(x))/根号下D(x),求E(X*),D(X*)

题目

设随机变量X的期望为E(x),方差为D(x)>0,X*=(X-E(x))/根号下D(x),求E(X*),D(X*)

答案

E(X*) = E[(X-E(X))/√D(X)] = [E(X)-E(X)]/√D(X) = 0D(X*) = E{[X*-E(X*)]^2} = E{[(X-E(x))/√D(x)]^2}= E[X-E(x)]^2] / D(X) = D(X)/D(X) = 1因此随机变量X*的数学期望E(X*) = 0,方差D(X*) = 1.