Poisson Distribution

Poisson distribution: counting rare events

The Poisson distribution counts how many events land in a fixed interval of time, space or volume, assuming those events are independent and arrive at a steady average rate λ: P(X = k) = e^(−λ)·λ^k / k!. One quirk stands out here: mean equals variance equals λ. It also stands in for the binomial(n, p) when n is large and p is small, taking λ = n·p. Say a bakery serves 3 customers per minute on average. The chance of seeing exactly 5 in some minute works out to e^(−3)·3^5 / 120 ≈ 0.1008, around 10.08%. Siméon Poisson introduced it in 1837, and Bortkiewicz later made it famous in 1898 by fitting it to deaths by horse kicks in the Prussian army.

Applications

Think of queueing theory, with M/M/1 systems in telecom and call centers, or radioactive decay, server failures and request arrivals. It also covers mutations per gene in genomics, defects per square metre of fabric in quality control, and goals per match in football, which average around 1.5 in the Premier League.

FAQ

When can I use Poisson instead of binomial? When n is large, usually n > 50, and p is small, around p < 0.1, so that λ = n·p stays moderate. The bigger n gets, the closer the approximation lands.

Why does mean equal variance? It's baked into the structure of the distribution. So if your data shows variance well above the mean (overdispersion), Poisson is simply the wrong model and you'd want the negative binomial instead.

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

What if events are not independent? Then the model falls apart. Bursts of clustered arrivals call for a Cox process or a negative binomial, and counts that are correlated may need a Hawkes process.