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The Poisson distribution

Peter Ralph

1 December 2020 – Advanced Biological Statistics

Stochastic minute

The Poisson distribution

If \(N \sim \Poisson(\mu)\) then \(N \ge 0\) and \[\begin{aligned} \P\{N = k\} = \frac{\mu^k}{k!} e^{-\mu} \end{aligned}\]

  • \(N\) is a nonnegative integer (i.e., a count)

  • \(\E[N] = \var[N] = \mu\)

  • If a machine makes widgets very fast, producing on average one broken widget per minute (and many good ones), each breaking independent of the others, then the number of broken widgets in \(\mu\) minutes is \(\Poisson(\mu)\).

  • If busses arrive randomly every \(\Exp(1)\) minutes, then the number of busses to arrive in \(\mu\) minutes is \(\Poisson(\mu)\).

Important point:

  • the standard deviation is the square root of the mean

plot of chunk r plot_poisson

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