Poisson distribution

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Story

The number \(n\) of arrivals of a Poisson process in unit time is Poisson distributed.

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Example

The number of mutations in a strand of DNA per unit length (since mutations are rare) are Poisson distributed.

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Parameters

The single parameter is the rate \(\lambda\) of the arrival of the Poisson process.

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Support

The Poisson distribution is supported on the set of nonnegative integers.

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Probability mass function

\[\begin{align} f(n;\lambda) = \frac{\lambda^n}{n!}\,\mathrm{e}^{-\lambda}. \end{align}\]

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Moments

Mean: \(\lambda\)

Variance: \(\lambda\)

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Usage

Package	Syntax
NumPy	rg.poisson(lam)
SciPy	scipy.stats.poisson(lam)
Stan	poisson(lam)

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Related distributions

• In the limit of \(N\to\infty\) and \(\theta\to 0\) such that the quantity \(N\theta\) is fixed, the Binomial distribution becomes a Poisson distribution with parameter \(\lambda = N\theta\). Thus, for large \(N\) and small \(\theta\), the Binomial distribution is well-approximating by the Poisson distribution. Considering the biological example of mutations, this is Binomially distributed: There are \(N\) bases, each with a probability \(\theta\) of mutation, so the number of mutations, \(n\) is Binomially distributed. Since \(\theta\) is small and \(N\) is large, it is approximately Poisson distributed.

• Under the \((\mu,\phi)\) parametrization of the Negative Binomial distribution, taking the limit of large \(\phi\) yields the Poisson distribution.

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PMF and CDF plots

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Links

• Wikipedia

• Numpy

• Scipy

• Stan