The Poisson Distribution

The Poisson distribution models the number of rare, independent events that occur in a fixed interval of time or space: new disease cases reported per week, mutations per genome, radioactive decays per second, or calls arriving at a help line. It is the go-to model for count data when events happen at a steady average rate.

Figure 1. The Poisson(3) pmf, where the mean equals the variance equals lambda.

Definition#

Let $X\sim\mathrm{Poisson}(\lambda)$ count events in a fixed window with average rate $\lambda$ . Its probability mass function is $P(X=k)=\frac{\lambda^{k}e^{-\lambda}}{k!}.$

Support: $k\in\{0,1,2,\dots\}$ .
Parameter: rate $\lambda>0$ (the expected number of events in the window).
Mean: $\mathbb{E}[X]=\lambda$ .
Variance: $\mathrm{Var}(X)=\lambda$ .

A striking feature is that the mean and variance are equal, both $\lambda$ .

Show that the mean and variance both equal

\lambda

Start from the definition $\mathbb{E}[X] = \sum_{k=0}^{\infty} k\,\frac{\lambda^k e^{-\lambda}}{k!}$ . The $k = 0$ term vanishes, and for $k \ge 1$ we can write $k/k! = 1/(k-1)!$ :

$\mathbb{E}[X] = \lambda e^{-\lambda}\sum_{k=1}^{\infty}\frac{\lambda^{k-1}}{(k-1)!} = \lambda e^{-\lambda}\,e^{\lambda} = \lambda ,$

since the shifted sum is just the Taylor series for $e^{\lambda}$ . For the variance, the same reindexing on $\mathbb{E}[X(X-1)]$ gives

$\mathbb{E}[X(X-1)] = \sum_{k=2}^{\infty} k(k-1)\frac{\lambda^k e^{-\lambda}}{k!} = \lambda^2 e^{-\lambda}\sum_{k=2}^{\infty}\frac{\lambda^{k-2}}{(k-2)!} = \lambda^2 .$

Then $\mathbb{E}[X^2] = \mathbb{E}[X(X-1)] + \mathbb{E}[X] = \lambda^2 + \lambda$ , so

$\operatorname{Var}(X) = \mathbb{E}[X^2] - \big(\mathbb{E}[X]\big)^2 = (\lambda^2 + \lambda) - \lambda^2 = \lambda .$

Mean and variance coincide — the signature of the Poisson.

Real count data with variance much larger than the mean are called overdispersed and signal that a plain Poisson model is too simple.

Limit of the binomial#

The Poisson arises as the limit of a binomial with many trials, each individually unlikely. If $n\to\infty$ and $p\to 0$ while the product $np\to\lambda$ stays fixed, then $\binom{n}{k}p^k(1-p)^{n-k}\;\longrightarrow\;\frac{\lambda^{k}e^{-\lambda}}{k!}.$ This is why the Poisson is called the “law of rare events”: it counts many opportunities for an event, each with tiny probability.

When it arises#

The Poisson applies to counts of independent events at a constant rate: epidemiological case counts, incidence of rare diseases, defects per batch, or arrivals in a queue. It connects directly to the exponential distribution: if event counts are Poisson, the waiting times between consecutive events are exponential.

In code#

R#

# pmf, cdf, quantile, and sampling for Poisson(lambda = 4)
dpois(2, lambda = 4)    # P(X = 2)
ppois(2, lambda = 4)    # P(X <= 2)
qpois(0.95, lambda = 4) # 95% quantile

set.seed(123)
x <- rpois(10000, lambda = 4)  # random sample
hist(x, breaks = seq(-0.5, max(x) + 0.5, 1), freq = FALSE)  # histogram
points(0:15, dpois(0:15, 4))                                # overlay the pmf

Python#

Python

import numpy as np
from scipy import stats

lam = 4
stats.poisson.pmf(2, lam)    # P(X = 2)
stats.poisson.cdf(2, lam)    # P(X <= 2)
stats.poisson.ppf(0.95, lam) # 95% quantile

rng = np.random.default_rng(123)
x = rng.poisson(lam, size=10000)  # random sample
# plt.hist(x, bins=range(0, 16), density=True); overlay stats.poisson.pmf(range(16), lam)

Julia#

Julia

using Distributions, Random

d = Poisson(4)      # Poisson(lambda)
pdf(d, 2)           # P(X = 2)   (pdf = pmf for discrete)
cdf(d, 2)           # P(X <= 2)
quantile(d, 0.95)   # 95% quantile

Random.seed!(123)
x = rand(d, 10_000) # random sample
# histogram(x, normalize=:pdf); scatter!(0:15, pdf.(d, 0:15)) to overlay the pmf

Simulation#

Many Poisson draws have empirical mean and variance both close to $\lambda$ — the signature equality of the distribution.

set.seed(7)
x <- rpois(1e6, lambda = 4)
mean(x)  # ~ 4.00  (theoretical mean = lambda)
var(x)   # ~ 4.00  (theoretical variance = lambda; mean == variance)

Why it matters for statistics#

The Poisson is the foundation of count-data modeling, including Poisson regression for rates (cases per person-year) and the analysis of contingency tables. Recognizing when the mean-equals-variance assumption fails — overdispersion — guides the choice of richer models such as the negative binomial. It links proportions (binomial) and waiting times (exponential) into one coherent picture of event processes.