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The Expectation and Variance of Poisson Random Variable

卜瓦松隨機變數的期望值:
$\begin{array}{rcl} E[X]&=& \displaystyle\sum_{i=0}^\infty\frac{ie^{-\lambda}\la... ...displaystyle\sum_{j=0}^\infty \frac{\lambda^j}{j!}=e^\lambda \\ \\ \end{array}$
卜瓦松隨機變數的變異數: 先求 E[X²]
$\begin{array}{rcl} E[X^2]&=&\displaystyle\sum_{i=0}^\infty\frac{i^2e^{-\lambda}... ...{-\lambda}\lambda^j}{j!}\right ] \\ \\ &=&\lambda (\lambda +1) \\ \end{array}$
所以, $Var(X)=E[X^2]-(E[X])^2=\lambda (\lambda +1)-\lambda ^2=\lambda$

Computing the Poisson Distribution Function

若 X 為參數是 $\lambda$ 的卜瓦松隨機變數, 則 $\displaystyle\frac{P\{X=i+1\}}{P\{X=i\}}= \frac{e^{-\lambda}\lambda^{i+1}/(i+1)!}{e^{-\lambda}\lambda^i/i!}$

從 $P\{X=0\}=e^{-\lambda}$ 開始, 我們可利用上式連續計算

$\begin{array}{rcl} P\{X=1\}&=& \lambda P\{X=0\} \\ P\{X=2\}&=& \displaystyle\f... ...dots& \\ P\{X=i+1\}&=& \displaystyle\frac{\lambda }{i+1} P\{X=i\} \end{array}$