Mathematics.

probability distributions

Poisson Distribution

Statistics25 minDifficulty3 out of 10

You should know: binomial distribution

Overview

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given that events happen independently and at a constant average rate λ (lambda). It's used for counts like the number of customer arrivals per hour, typos per page, or radioactive decays per second. Unlike the binomial distribution, the Poisson has a single parameter λ, which equals both its mean and its variance. It arises as the limiting case of the binomial distribution when the number of trials n grows large and the success probability p shrinks small, with np held constant at λ.

Intuition

Think of splitting a time interval into many tiny sub-intervals, each so short that at most one event can plausibly occur in it. Each sub-interval is like a Bernoulli trial with a very small success probability, and there are a huge number of them — exactly the setup that produces a binomial distribution in the np → λ limit. The Poisson distribution is the mathematical shortcut for this limit, letting you skip tracking individual sub-intervals and work directly with the overall rate λ.

Formal Definition

Definition

If X counts the number of events in a fixed interval with average rate λ, X ~ Poisson(λ), with probability mass function:

P(X=k)=λke−λk!,k=0,1,2,…P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \qquad k = 0, 1, 2, \ldots
Poisson PMF
E[X]=λE[X] = \lambda
Mean
Var⁡(X)=λ\operatorname{Var}(X) = \lambda
Variance (equals the mean)

Worked Examples

  1. Apply the Poisson PMF with λ=3, k=2.

    P(X=2)=32e−32!=9e−32P(X=2) = \frac{3^2 e^{-3}}{2!} = \frac{9 e^{-3}}{2}

  2. Numerically, e^{-3} ≈ 0.0498, so compute the value.

    P(X=2)≈9×0.04982≈0.4482≈0.224P(X=2) \approx \frac{9 \times 0.0498}{2} \approx \frac{0.448}{2} \approx 0.224

Answer: P(X = 2) ≈ 0.224.

Practice Problems

Difficulty 3/10

For X ~ Poisson(λ = 2), find P(X = 1).

Difficulty 3/10

For X ~ Poisson(λ = 5), find the mean and variance.

Difficulty 5/10

A factory averages 1 machine breakdown per week (λ = 1). Find the probability of at least 1 breakdown in a given week.

Quiz

The Poisson distribution arises as a limiting case of the binomial distribution when:
For a Poisson(λ) random variable, the variance is:
Which scenario is best modeled by a Poisson distribution?

Summary

  • The Poisson distribution models counts of independent events occurring at a constant average rate λ over a fixed interval.
  • Its PMF is P(X=k) = λᵏe⁻λ/k!, with mean and variance both equal to λ.
  • It arises as the limit of the binomial distribution as n → ∞, p → 0, with np = λ held fixed.

References

  1. WebsiteWikipedia — Poisson distribution