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Poisson Distribution

Formula

\[ X\sim \mathrm{Poisson}(\lambda),\qquad P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!},\quad k=0,1,2,\dots \]
\[ \mathbb{E}[X]=\lambda,\qquad \operatorname{Var}(X)=\lambda \]

Plot

type: bars
xs: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
ys: 0.13534 | 0.27067 | 0.27067 | 0.18045 | 0.09022 | 0.03609 | 0.01203 | 0.00344
xmin: -0.5
xmax: 7.5
ymin: 0
ymax: 0.31
height: 280
title: Poisson PMF example (lambda=2)

Parameters

  • \(\lambda>0\): average count/rate in a fixed interval
  • \(k\): nonnegative integer count

What it means

Models the number of events occurring in a fixed interval when events happen independently at an approximately constant rate.

What it's used for

  • Count data (arrivals, defects, clicks per interval).
  • Rare-event approximations to binomial counts.

Key properties

  • Discrete distribution on nonnegative integers.
  • Mean equals variance (\(\lambda\)).

Common gotchas

  • "Poison" is a common typo; the distribution is "Poisson."
  • If variance is much larger than mean, a plain Poisson model may fit poorly (overdispersion).

Example

If \(\lambda=2\), then \(P(X=0)=e^{-2}\approx 0.135\).

How to Compute (Pseudocode)

Input: distribution parameters and query values (for PMF/PDF/CDF or moments)
Output: distribution quantities

validate parameters
for each query value x (or count k):
  evaluate the PMF/PDF/CDF formula from the card
optionally compute moments/statistics from known closed forms or by summation/integration
return the requested values

Complexity

  • Time: Typically \(O(q)\) for \(q\) query values once parameters are known (assuming constant-time formula evaluation per query)
  • Space: \(O(q)\) for output values (or \(O(1)\) for a single query)
  • Assumptions: Parameter estimation/fitting cost is excluded; numerical special-function evaluation can affect constants for some families

See also