Poisson Distribution

See also: Poisson Limit Theorem

def. Poisson Distribution. A random variable $X$ which model the number of events in a fixed interval of time, where each event is rare and there is a large number of events, is well modeled by a Poisson Distribution with the intensity of the event $\lambda$:

$$\begin{gathered} X\sim \text{Poisson}(\lambda ) \\ P(X=k)=e^{-\lambda }\frac{{\lambda }^k}{k!} \end{gathered}$$

• $\mathbb{E}(X)=\lambda$
• $\mathbb{SD}(X)=\sqrt{\lambda }$
• $\mathbb{P}(X<n)={\sum }_{k=0}^{n-1}{e}^{-\lambda }\frac{{\lambda }^k}{k!}$

Estimators §

let $X_1,\dots ,X_n\sim \text{Poi}(\lambda )$

$$\ln {\mathcal{L}}_n=\frac{{\lambda }^{{\sum }_{i=1}^n{x}_i}{e}^{n\lambda }}{{\prod }_{i=1}{x}_i!}$$ $$s_n=\frac{{\sum }_{i=0}^n{x}_i}{\lambda }-n$$ $${\hat{\lambda }}_{\text{MLE}}=\frac{{\sum }_{i=1}^n{x}_i}{n}={\bar{X}}_n$$

single	multiple
$I=\frac{1}{\lambda }$	$I_n=\frac{n}{\lambda }$