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！