Poisson distribution

When events are rare but observed over a long interval of time or space (calls to a switchboard, decays, typos in a book), the suitable model is the Poisson distribution.

Property — $X\sim\text{Pois}(\lambda)$

It models the number of rare events in an interval of time/space. $\lambda$ is the mean number of events. $P(X=k) = e^{-\lambda}\frac{\lambda^k}{k!}, \qquad k=0,1,2,\ldots$ Mean = variance = $\lambda$.

A distinctive feature: in the Poisson, mean and variance coincide, both equal to $\lambda$. The graph that follows shows the case $\lambda=4$ (for example $4$ calls per hour on average).

Poisson distribution with $\lambda=4$: the peak is around $k=3$ and $k=4$, and the tail stretches to the right.

Links

Topics: Probability distributions
Concepts: Poisson distribution · Expected value · Variance
Skills: Probability calculation · Modelling
People: Siméon-Denis Poisson