Poission Distribution

STAT1003 – Statistical Techniques

Dr. Emi Tanaka

Australian National University

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Poisson distribution

  • If \(X \sim B(n, p)\) and \(n \to \infty\), \(p \to 0\) such that \(np = \lambda\) (a constant), then

    \[ P(X = k) = \lim_{n \to \infty} \binom{n}{k} p^k (1 - p)^{n - k} = \frac{e^{-\lambda} \lambda^k}{k!}. \]

  • This limiting distribution is called the Poisson distribution with parameter \(\lambda\) written as \(X \sim \mathrm{Poisson}(\lambda)\).

  • \(E(X) = \lambda\) and \(\mathrm{Var}(X) = \lambda\).

Poisson distribution in practice

  • The Poisson distribution is often used to model the number of times a discrete event occurs within a given, fixed interval of time, distance, area, or volume.
  • For example, it can be used to model:
    • the number of emails received per hour, or
    • the number of accidents at an intersection per month.
  • The parameter \(\lambda\) is the expected number of occurrences in the interval.

If a coffee shop has an average of 75 customers per hour then assume the number of customers \(X \sim \mathrm{Poisson}(75)\). Then probability of exactly 70 customers arriving in an hour is \[ P(X=70) = \frac{75^{70}e^{-75}}{70!} \approx 0.0402. \]

Poisson distribution in R

  • Suppose \(X \sim \mathrm{Poisson}(75)\).
  • Then \(P(X = 70)\):
  • Then \(P(X \leq 70)\):
  • To simulate 1000 random variables from \(X \sim \mathrm{Poisson}(75)\):

Summary

\[X \sim \mathrm{Poisson}(\lambda)\]

  • \(P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\) for \(k = 0, 1, 2, \ldots\).
  • \(E(X) = \lambda\) and \(\mathrm{Var}(X) = \lambda\).
  • Poisson distribution is the limiting distribution of the Binomial distribution when \(n \to \infty\), \(p \to 0\) such that \(np = \lambda\) (a constant).
  • Poisson distribution is often used to model the number of times a discrete event occurs within a given, fixed interval of time, distance, area, or volume.