求概率论题解答题目如下，求大神LetX1andX2betwoindependentstandardnormaldistributedrandomvariables.LetUbeindependentofX1andX2andisuniformlydistributedon(0,1).DefineZ=UX1+(1−U)X2.Findthemeanandv

Question

求概率论题解答
题目如下，求大神 Let X1 and X2 be two independent standard normal distributed random variables.
Let U be independent of X1 and X2 and is uniformly distributed on (0, 1). Define
Z = UX1 + (1 − U)X2. Find the mean and variance of Z.

Answer 1

E(Z)=E( UX1 + (1 − U)X2)=E( U)E(X1) + E((1 − U))E(X2) = 0V(Z)=E(Z^2)-(E(Z)^2)=E(Z^2)=E(U^2 X1^2 + (1-U)^2(X2)^2 + 2U(1-U)X1X2)=E(U^2)*1+E((1-U)^2)*1+0=2E(U^2) =2 积分（0 到 1) x^2 dx = 2/3...