X服从泊松分布求E[X(X-1)]

设X服从泊松分布，参数为λ，那么
EX=λ, DX=λ，

所以 E[X(X-1)]
=E(X^2)-EX
=DX+(EX)^2-EX
=λ＋λ^2－λ
=λ^2.

也可以直接根据定义
E[X(X-1)]
=sum(n(n-1)*λ^n/n!*e^(-λ)), n=0..∞
=sum(λ^2*λ^(n-2)/(n-2)!*e^(-λ)), n=2..∞
=λ^2*sum(λ^n/n!*e^(-λ)), n=0..∞
=λ^2*1
=λ^2