Mathematics and Statistics · Ch 16 — Probability Distributions

Poisson Distribution

The Poisson distribution models the number of times a rare event happens over a fixed interval of time, length, area or volume — the number of accidents at a junction per week, misprints per page, telephone calls per minute, or defective items in a large batch. Unlike the binomial, there is no fixed $n$ ; instead we know only the average number of occurrences.

Note

Poisson probability formula

A discrete variable $X$ taking values $0, 1, 2, \ldots$ follows a Poisson distribution with parameter $m > 0$ (written $X \sim P(m)$ ) if

$P(X = r) = \frac{e^{-m}\, m^{r}}{r!}, \qquad r = 0, 1, 2, \ldots$

where $m$ is the average (expected) number of occurrences and $e \approx 2.71828$ .

The values run over all non-negative integers (there is no upper limit), and they add to $1$ because $\displaystyle\sum_{r=0}^{\infty} \frac{m^r}{r!} = e^{m}$ , so $\sum_r P(X=r) = e^{-m}\,e^{m} = 1$ .

Note

Mean and variance of the Poisson distribution

$\text{Mean} = E(X) = m, \qquad \operatorname{Var}(X) = m.$

A defining feature of the Poisson distribution is that its mean and variance are equal, both equal to $m$ .

Tip

Poisson as a limit of the binomial

When $n$ is large and $p$ is small (a rare event over many trials), the binomial $B(n,p)$ is closely approximated by the Poisson distribution with $m = np$ . This is why Poisson is the natural model for "few defectives in a large sample" problems: take $m = np$ and use the Poisson formula. …

Definition 12Poisson distribution $P(m)$

For rare events with average count $m$ : $P(X=r)=\dfrac{e^{-m}m^r}{r!}$ , $r=0,1,2,\ldots$ ; both mean and …

Definition 13Poisson approximation to the binomial

For large $n$ and small $p$ , $B(n,p)$ is approximated by the Poisson distribution with p …

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