The Poisson distribution and rare events

What you’ll learn

• Model event counts with a rate and interval.
• Calculate Poisson probabilities and combine intervals.
• Use a Poisson approximation to a binomial distribution.

Understand the idea

A Poisson model describes a count of events over a specified interval of time, length, area or volume. Its parameter $\lambda$ is the expected count in that interval. A useful process model has independent occurrences, a constant average rate and negligible chance of multiple events in a sufficiently small interval.

Scale the parameter when the interval changes. If an average of 2.5 events occurs per hour, the expected count in four hours is 10. Independent Poisson counts add to a Poisson count whose parameter is the sum of their parameters. The equality of mean and variance is a feature to assess when judging the model.

For a large number of independent trials with a small success probability, $B(n,p)$ can be approximated by a Poisson distribution with $\lambda=np$. State that the result is approximate. A clustered arrival process or a changing event rate can make a Poisson model unsuitable.

Key formulae

$$P(X=r)=\frac{e^{-\lambda}\lambda^r}{r!}$$

$$E(X)=\operatorname{Var}(X)=\lambda$$

$$X+Y\sim\operatorname{Po}(\lambda+\mu)\quad\text{for independent Poisson variables}$$

$$B(n,p)\approx\operatorname{Po}(np)$$

Worked example

Calls arrive according to a Poisson model with mean 2 per five-minute period. Find the probability of at most one call in a five-minute period.

1. For this interval, $X\sim\operatorname{Po}(2)$.
2. At most one means $P(X=0)+P(X=1)$.
3. This is $e^{-2}+2e^{-2}=3e^{-2}=0.4060$ approximately.

Answer: $0.4060$ approximately.