Poisson Distribution

The Poisson distribution models the count of independent events in a fixed interval when they occur at a steady average rate.

The model

A Poisson(λ) random variable counts events in a window when they arrive independently at an average rate λ per window. Its PMF is P(X = k) = e^{−λ} λᵏ / k! for k = 0, 1, 2, …. The single parameter λ is both the mean and the variance.

Equal mean and variance

E[X] = Var(X) = λ is a distinctive fingerprint. If observed counts show variance much larger than the mean, the Poisson assumption is violated — usually by clustering or a varying rate — a condition called overdispersion that calls for a different model.

Relationship to other distributions

• It is the limit of Binomial(n, p) as n → ∞ with np → λ (rare events, many trials).
• The gaps between Poisson events follow an exponential distribution.
• For large λ it is well approximated by a normal with mean and variance λ.

Counting applications

Poisson statistics govern counting detectors. The number of neutrons or photons registered in a fixed interval, when the source rate is steady and events are independent, is Poisson. This is why counting uncertainty scales as √N: the standard deviation of a Poisson count is √λ, so relative uncertainty is 1/√λ.

from math import exp, factorial
def poisson_pmf(k, lam):
    return exp(-lam)*lam**k/factorial(k)
print(round(poisson_pmf(2, 3.0), 4))  # 0.2240

Because relative precision improves only as √N, a counting measurement that needs 1% precision requires on the order of ten thousand counts.