Section 7.3

The Poisson Distribution

Poisson probabilities, thinning and superposition of Poisson counts, and Poisson limits such as fixed points of a random map.

Toolkit

A Poisson process or Poisson random variable models counts of rare, independent events; it is the limit of $\mathrm{Bin}(n,p)$ when $n\to\infty$ , $p\to0$ , $np\to\lambda$ .

\mathbb{P}(N=k)=\frac{\lambda^k e^{-\lambda}}{k!},\qquad \mathbb{E}[N]=\operatorname{Var}(N)=\lambda.

Key tools:

Thinning: if each of a $\mathrm{Poisson}(\lambda)$ number of points is independently kept with probability $p$ , the kept count is $\mathrm{Poisson}(\lambda p)$ .
Superposition: independent $\mathrm{Poisson}(\lambda_1),\dots$ sum to $\mathrm{Poisson}(\sum\lambda_i)$ .
Time-scaling: for a rate- $\lambda$ process, counts in disjoint intervals are independent Poisson with mean $\lambda\times(\text{length})$ ; no-event probability over length $t$ is $e^{-\lambda t}$ .
Mode: ratio $\mathbb{P}(N=k)/\mathbb{P}(N=k-1)=\lambda/k$ , so the pmf increases while $k<\lambda$ and decreases once $k>\lambda$ .
Poisson limit theorems: sums of many weak, nearly independent indicator contributions (e.g. rare matches, fixed points) converge to Poisson with $\lambda$ equal to the limiting expected count.

Problems