求泊松随机变量期望值公式的证明,还有方差,母函数的公式证明,望各位指教,

泊松分布：P{X=k}=λ^k·exp(-λ)/k!(k=0,1,2,...λ>0)
E(X)=∑kP{X=k}=∑λ^k·exp(-λ)/(k-1)!(k从1到+无穷)
由泰勒展式有
exp(λ)=∑λ^k/k!(k从0到+无穷)=∑λ^(k-1)/(k-1)!(k从1到+无穷)
代入上式得
E(X)=λexp(λ)exp(-λ)=λ
E(X^2)=∑k^2·P{X=k}=∑k·λ^k·exp(-λ)/(k-1)!(k从0到+无穷)
=∑(k-1)·λ^k·exp(-λ)/(k-1)!+∑λ^k·exp(-λ)/(k-1)!
=∑(k-1)·λ^k·exp(-λ)/(k-1)!+λ (k从0到+无穷)
=∑λ^k·exp(-λ)/(k-2)!+λ
=λ^2+λ
D(X)=E(X^2)-[E(X)]^2=λ
母函数啥?