Question

E(X^2)的期望如何求解？

Answer 1

E(X^2) 是 X^2 的期望.

比如，P{X=1} = 2/3, P{X=0} = 1/6, P{X=-1} = 1/6.

EX = 1 * 2/3 + 0 * 1/6 +(-1) * 1/6 = 2/3 - 1/6 = 1/2.
EX^2 = 1^2 * 2/3 + 0^2 * 1/6 + (-1)^2 * 1/6 = 2/3 + 1/6 = 5/6.
DX = EX^2 - [EX]^2 = 5/6 - (1/2)^2 = 7/12

以上内来源于： 方差计算公式D(X) = E(X^2) - [E(X)]^2

但是根据期望的定义：EX=累计所有的P(Xi)*Xi

所以E(X^2) = 累加P(Xi^2) * Xi^2

本题P(X^2 = 1) = P(-1^2 = 1)+P(1^2 = 1) = 5/6, P(X^2 = 0) = 1/6

所以E(X^2) = 5/6 * 1+1/6 * 0 = 5/6

若取Y = X^2,则更好理解，因为Y的取值只有1和0