The Expectation and Variance of Poisson Random Variable

Computing the Poisson Distribution Function

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

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

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