如何证明离散型随机变量D(aX+b)=a^2D(X)

Question

如何证明离散型随机变量D(aX+b)=a^2D(X)

Answer 1

D(aX+b)=E[aX+b-E(aX+b)]^2
=E[aX+b-E(aX)-E(b)]^2
=E[aX+b-aE(X)-b]^2
=E{a[X-E(X)]}^2
=E{a^2[X-E(X)]^2}
=a^2E[X-E(X)]^2
=a^2D(X)
证毕