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Poisson distribution

What it describes

The Poisson distribution counts events in an interval when they occur independently, at constant average rate, with negligible simultaneity. One parameter lambda sets both mean and variance.

Support is {0,1,2,…}. lambda is the expected count in the chosen interval, not a probability between 0 and 1.

History and names

Siméon-Denis Poisson presented the law in 1837 while studying probabilities of legal judgements and rare events. Modern theory connects it to arrivals and the Binomial limit.

How it connects to other distributions

It is the rare-event limit of Binomial and the count companion of Exponential waiting times. The difference of two independent Poisson variables is Skellam; Gamma-mixing the rate gives Negative Binomial.

Where it appears

  • arrivals, calls, failures, or accidents per unit time
  • rare-event counts in traffic, epidemiology, and astronomy

Fitting cautions

Mean-equals-variance is a model implication, not a data law; checking it alone does not validate independence or constant rate.

When a rate becomes a count

Poisson published his law in 1837, but its applied reputation grew through examples of rare events. In 1898 Bortkiewicz studied deaths from horse kicks across Prussian army corps, a historical case showing how scattered counts could be compared through a common rate.

The model has a demanding signature: its mean equals its variance. Extra variance may reveal heterogeneous rates, contagion, or clustered events; too little may indicate inhibition or a limited number of opportunities. Adding independent Poisson processes adds their rates. Conditioning two independent counts on their total makes either count Binomial. These properties make Poisson a central piece of queueing, epidemiology, and point processes, not merely a curve for small numbers.

Decision guide

A good candidate when: independent events are counted over known exposure with an approximately constant rate and mean close to variance.

Compare it with: Negative Binomial for overdispersion and models with exposure, trends, or zero inflation when appropriate. Equality of mean and variance is a diagnostic, not a constraint to force.

References

  • SciPy reference: scipy.stats.poisson — definition and parameterization

  • Bortkiewicz, L. von (1898). Das Gesetz der kleinen Zahlen. Teubner.

  • Johnson, N. L., Kemp, A. W. & Kotz, S. (2005). Univariate Discrete Distributions, 3rd ed. Wiley.

  • Poisson, S.-D. (1837). Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Bachelier.

  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.

Poisson Distribution: equations and calculator

Distribution defintion

X∼Poisson(λ)X\sim\mathrm{Poisson}\left(\lambda\right)

Distribution domain

x∈N≡{0,1,2,… }x\in\mathbb{N}\equiv \left\{0,1,2,\dots\right\}

Parameters domain and parameters constraints

λ∈R+\lambda\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=e−λ∑i=0xλii!=1−γ(x+1,λ)x!=1−P(x−1,λ)F_{X}\left(x\right)=e^{-\lambda} \sum_{i=0}^{x} \frac{\lambda^i}{i!}=1-\frac{\gamma(x+1, \lambda)}{x!}=1-P(x-1,\lambda)

Probability mass function

fX(x)=λxe−λx!f_{X}\left(x\right)=\frac{\lambda^x e^{-\lambda}}{x!}

Percent point function/Sample

FX−1(u)=arg⁡min⁡x∣FX(x)−u∣F^{-1}_{X}\left(u\right)=\arg\min_{x}\left| F_{X}\left(x\right)-u \right|

Parametric centered moments

E[Xk]=μk′=∑x=0∞xkfX(x)E[X^k]=\mu'_{k}=\sum_{x=0}^{\infty}x^{k}f_{X}\left(x\right)

Parametric mean

Mean(X)=μ1′=λ\mathrm{Mean}(X)=\mu'_{1}=\lambda

Parametric variance

Variance(X)=(μ2′−μ1′2)=λ\mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=\lambda

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=λ−1/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\lambda^{-1/2}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+λ−1\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3+\lambda^{-1}

Parametric median

Median(X)=⌊λ+1/3−0.02/λ⌋\mathrm{Median}(X)=\lfloor\lambda+1/3-0.02/\lambda\rfloor

Parametric mode

Mode(X)=⌊λ⌋\mathrm{Mode}(X)=\lfloor\lambda\rfloor

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. However, it is possible to calculate the Percentile Point Function by approximating it to the nearest integer.
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
⌊x⌋:Floor function\lfloor{x}\rfloor: \text{Floor function}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma functionP\left(a,x \right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
γ(a,x):Lower incomplete Gamma function\gamma\left(a,x \right):\text{Lower incomplete Gamma function}