Poisson Distribution¶

Definition¶

Let $\lambda>0$$0$. A discrete random variable $X$ has a Poisson distribution with parameter $\lambda$, written $X\sim\operatorname{Pois}(\lambda)$$\mathrm{Pois}(\lambda )$, if, for $k=0,1,2,\ldots$$0,1,2,\dots$,

and the PMF is zero elsewhere. The extension $\lambda=0$ gives the constant-zero distribution.

Interpretation¶

A Poisson variable models a count of events in a fixed period, such as arrivals or claims, when there is no fixed upper limit on the count. Its single parameter $\lambda$ is the mean count.

The figure shows $\operatorname{Pois}(3)$. The bars at $2$ and $3$ are equally high, $\tfrac{9}{2}\mathrm{e}^{-3}\approx0.224$$0.224$, and the bars fall off quickly on both sides: the values above $10$ have total probability below $0.001$. The mean $\lambda=3$ is marked.

The ratio of neighbouring masses explains the shape: $\mathbb{P}(X=k)/\mathbb{P}(X=k-1)=\lambda/k$$\lambda /k$, so the bars increase while $k<\lambda$ and decrease once $k>\lambda$$\lambda$. When $\lambda$ is an integer, as here, the two bars at $\lambda-1$ and $\lambda$ are equal. The factorial makes the masses eventually shrink faster than any geometric sequence, and they add up to $1$ because $\displaystyle \sum_{k\ge0}\lambda^k/k!=\mathrm{e}^{\lambda}$${\mathrm{e}}^{\lambda }$, the series of the exponential function.

The Poisson distribution approximates the binomial distribution with many trials and a small success probability: for large $n$, $\operatorname{Bin}(n,\lambda/n)$ is close to $\operatorname{Pois}(\lambda)$, as the Further Properties state. This is why it fits counts of rare events among many independent opportunities. The same distribution describes the number of arrivals in an interval of a Poisson process.

The parameter is also the variance, so a Poisson model fixes the spread once the mean is known. Count data whose variance is clearly larger than the mean are therefore not well described by a single Poisson distribution.

Key Results¶

• $\mathbb{E}X=\operatorname{Var}(X)=\lambda$$\lambda$.
• MGF $M_X(t)=\exp(\lambda(\mathrm{e}^t-1))$ for every real $t$.

Examples¶

• For $\lambda=2$$2$, $\mathbb{P}(X=0)=\mathrm{e}^{-2}$ and $\mathbb{P}(X=1)=2\mathrm{e}^{-2}$$2{\mathrm{e}}^{-2}$.
• A Poisson count with mean $5$ has variance $5$ and standard deviation $\sqrt{5}$.

Further Properties¶

Independent Poisson counts add to a Poisson count with the sum of their parameters. A binomial model converges to this law when the number of trials tends to infinity and the success probabilities shrink so that their product tends to a fixed parameter.

Prerequisites¶

• Discrete Random Variable
• Factorial
• Exponential Function

Related Concepts¶

• Exponential Distribution
• Sum of Independent Random Variables
• Binomial Distribution
• Expectation
• Poisson Splitting
• Conditioning Trick for Variance
• Taylor Series
• Exponential Limit

Sources¶

• Probability Theory for EOR · Week 4
• Nicky van Foreest · Probability Distributions · Refresher

Aliases: Pois(lambda), Poisson random variable, Poisson r.v., Poisson variable Symbols: lambda, square root, sum

Last updated: 20 October 2026 03:02