概率论与数理统计题:证明:若X与Y相互独立,则D(X+Y))=D(X)+D(Y)
问题描述：
概率论与数理统计题:证明:若X与Y相互独立,则D(X+Y))=D(X)+D(Y)
答
设Z = X + Y
E(Z)=E(X)+E(Y)
方差的定义：D(Z) = E{(Z-E(Z))²}
D(Z) = D(X+Y) = E{(X+Y)² - (E(X)+E(Y))²} = E(X² - E²(X)) + E(Y² - E²(Y))+
+ E(2XY) - 2E(X) E(Y) = D(X) + D(Y) + 0
即：D(X+Y) = D(X) + D(Y)