Question

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)

Answer 1

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)