Question

设总体X~N(0,1),X1,X2,X3是来自X的样本,x为样本均值,则X3-x~___分布

要详细的解题步骤，谢谢

Answer 1

x=1/3(X1+X2+X3)
X3-x=X3-1/3(X1+X2+X3)=-1/3X1-1/3X2+2/3X3
所以E(X3-x)=-1/3EX1-1/3EX2+2/3EX3=0
Var(X3-x)=1/9Var(X1)+1/9Var(X2)+4/9Var(X3)=1/9+1/9+4/9=2/3
所以X3-x~N(0,2/3)