设随机变量x的是学期望为E(x),方差为D(x),证明对任意常数C,都有E(x-c)^2>=D(x)

Question

设随机变量x的是学期望为E(x),方差为D(x),证明对任意常数C,都有E(x-c)^2>=D(x)

Answer 1

E(x-c)²
= E(x²-2cx+c^2)
= E(x²)-2cE(x)+c² (1)
D(x) = E(x²)-E²(X) (2)
(1)-(2)：
E(x-c)²-D(x) =
= -2cE(x)+c²+E²(x)
= [E(x)-c]²>= 0 (3) 问题得证！