AHL 4.17 (HL)—Poisson distribution

A Poisson random variable counts events in a time, length or area interval when events occur independently at a stable average rate λ. Its mean and variance are both λ, and disjoint intervals have independent counts under the model.

If the interval is multiplied by k, the mean rate becomes kλ. Use the model only when events are countable, the rate is approximately constant and one event does not make another more or less likely.

If calls arrive at an average rate of 3 per hour, the count in two hours is Poisson with mean 6. The probability of exactly four calls uses that interval's λ, not the original one-hour value.

Poisson is not any small count. Clustering, a changing rate or a maximum capacity can violate the assumptions even when the data look discrete.

If independent counts satisfy $X\sim Pois(\lambda_1)$ and $Y\sim Pois(\lambda_2)$, then $X+Y\sim Pois(\lambda_1+\lambda_2)$. Choose binomial for successes in fixed independent trials with constant $p$, Poisson for independent events at a uniform average rate, and normal for an appropriate continuous symmetric measurement model. Match $\lambda$ to the interval before using technology.