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Poisson Distribution

NIST/SEMATECH Section 1.3.6.6.19 Poisson Distribution

PMF

Pois(l=5) PDF 0 2 4 6 8 10 12 14 k 0 0.05 0.1 0.15 P(X=k)

CDF

Pois(l=5) CDF 0 2 4 6 8 10 12 14 k 0 0.2 0.4 0.6 0.8 1 F(x)

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PMF

CDF

Formulas

Probability Mass Function

P(X=k)=λke−λk!,k=0,1,2,…P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \ldots

Cumulative Distribution Function

F(k)=e−λ∑i=0⌊k⌋λii!F(k) = e^{-\lambda} \sum_{i=0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!}

Properties

Mean

λ\lambda

Variance

λ\lambda

Overview

The Poisson distribution models the number of events occurring in a fixed interval of time or space when events occur independently at a constant average rate λ\lambda. It is used for count data in fields ranging from telecommunications to epidemiology.

Related Distributions

Binomial Distribution Exponential Distribution Normal Distribution