I Possion,Gaussion,Exponential distribution
II Ex,Ex^2,Dx prove
$-Possion\ distribution:$
$X~\pi(\lambda)$
$E(x)=\sum_{k=0}^{n}k\frac{\lambda^{k}e^{-\lambda}}{k!}$
$=\lambda e^{-\lambda}\sum^{\infty}_{(k-1)!}=\lambda e^{-\lambda}e^{\lambda}$
$=\lambda$
$E(X^{2})=E[X(X-1)+X]=E[X(x-1)]+E(X)$
$=\sum_{k=0}^{\infty}k(k-1)\frac{\lambda^{k}e^{-\lambda}}{k!}+\lambda$
$=\lambda^{2}e^{-\lambda}\sum_{k=2}^{\infty}\frac{e^{k-2}}{(k-2)!}+\lambda$
$=\lambda^{2}+\lambda$
$D(X)=\lambda^{2}$
$-Normality\ distribution:$
$-Exponential\ distribution:$
$E(X)=\int_{-\infty}^{\infty}x\frac{1}{\theta}e^{-\frac{x}{\theta}}dx$
$=-xe^{-\frac{x}{\theta}}|^{\infty}_{0}+\int_{0}^{\infty}e^{-\frac{x}{\theta}}dx$
$=\theta$
$E(X^{2})=\int_{-\infty}^{\infty}x^{2}\frac{1}{\theta}e^{\frac{x}{\theta}}dx$
$=2\theta^{2}$
$D(X)=\theta^{2}$