The Poisson Distribution

• X ~ Po(λ): P(X = r) = e^−λλ^r/r!, r = 0, 1, 2, …
• Mean = variance = λ.
• Model conditions: events occur singly, independently, at random, at a constant average rate.
• Change the interval by scaling λ in proportion.
• Po(np) approximates B(n, p) when n > 50 and np < 5.
• N(λ, λ) approximates Po(λ) when λ > 15, with a continuity correction.

Random events in an interval

Use the Poisson distribution for the number of events in a fixed interval of time or space when the events occur randomly, independently, singly and at a constant average rate. Explain suitability in context: "the calls arrive independently and at a constant average rate". A quick check from data: if the sample mean and variance are roughly equal, Poisson is plausible.

If the rate is 3 per hour, the number in half an hour is Po(1.5) and the number in 4 hours is Po(12).

When and how

Binomial → Poisson. If X ~ B(n, p) with n large and p small (roughly n > 50 and np < 5), then X ≈ Po(np). No continuity correction is needed: both are discrete.

Poisson → Normal. If λ is large (λ > 15), then X ≈ N(λ, λ). This is discrete to continuous, so use a continuity correction.