Poisson P(X=k)

Poisson P(X = k): probability mass function

The Poisson distribution tells you how likely it is that a given number of independent events land inside a fixed window (a span of time, an area, a volume) when those events show up at some steady average rate λ. The formula is P(X = k) = e^(−λ)·λ^k / k!. One thing sets it apart from most distributions, and it's worth remembering: its mean and variance are both equal to λ. When n is large and p is small, it also stands in nicely for the binomial(n, p), with λ = n·p. Take a call centre that fields 3 calls a minute on average. How likely is exactly 5 calls in some minute? You get e^(−3)·3^5 / 120 ≈ 0.1008, roughly 10%.

Applications

You'll find it in queueing theory (the M/M/1 model in call centres and telecom), in radioactive decay, and in counting defects per unit on a production line. It also shows up for goals per football match and mutations per gene in genomics. And then there's the old Bortkiewicz (1898) study, which counted deaths by horse kicks in the Prussian army — a textbook example to this day.

FAQ

When can Poisson replace the binomial? Once n gets large (n > 50) and p stays small (p < 0.1), so λ = n·p doesn't blow up. The bigger n gets, the closer the fit.

Why must mean equal variance? That equality is baked into how the distribution is built. When your sample variance runs well above the mean (what people call overdispersion), Poisson just isn't the right model, and the negative binomial is usually where you turn next.

Can k be a non-integer? No. Poisson is counting whole events, so k ∈ {0, 1, 2, …}. If what you actually care about is the continuous waiting time between events, reach for the exponential distribution instead.

What if k is bigger than λ? Out in the tail the probabilities drop off fast. When k sits far above λ you'll get a tiny number, but it never hits exactly zero. In principle, any count at all can happen.