Poisson Distribution

The Poisson distribution models the number of occurrences of a rare event in a fixed

interval when events happen independently at a constant average rate. If $X$ is

Poisson with parameter $\lambda>0$, then

$P(X=k)=\dfrac{e^{-\lambda}\lambda^{k}}{k!},\qquad k=0,1,2,\dots$

A defining feature is that the mean equals the variance, both equal to $\lambda$:

$E(X)=\operatorname{Var}(X)=\lambda$.

Typical problems: (i) read $\lambda$ from the stated average or variance; (ii) form

ratios such as $\dfrac{P(X=1)}{P(X=2)}=\dfrac{2}{\lambda}$ to solve for $\lambda$;

(iii) evaluate cumulative probabilities like

$P(X\ge1)=1-e^{-\lambda}$ or $P(X\le1)=e^{-\lambda}(1+\lambda)$. The Poisson also arises as

the limiting case of a binomial $B(n,p)$ when $n\to\infty,\ p\to0$ with $np=\lambda$

fixed, which is why binomial conditions (e.g. $P(X=1)=P(X=2)$) are sometimes used to …