When Is the Poisson Distribution an Appropriate Model?

A Poisson distribution is appropriate when counting independent events occurring at a constant average rate within a fixed interval of time or space. If these conditions hold, write $X \sim \operatorname{Po}(\lambda)$, where $\lambda$ is the expected number of events in that interval.

Conditions for a Poisson model

For $X \sim \operatorname{Po}(\lambda)$, both the mean and variance equal $\lambda$:

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

This equality can support the choice of model when analysing data, but it does not replace checking independence and a uniform rate.

Also distinguish Poisson from binomial: Poisson counts events in a continuous interval with no fixed number of trials, whereas binomial counts successes in a fixed number of independent trials with constant success probability. Choose the model from the context.

Scaling the parameter

Suppose a help desk receives an average of three calls per hour. For a 20-minute interval, the parameter must be scaled:

$\lambda=3\left(\frac{20}{60}\right)=1.$

Therefore, if $X$ is the number of calls in 20 minutes, $X \sim \operatorname{Po}(1)$. The probability of at least one call is

$P(X\geq 1)=1-P(X=0)=1-e^{-1}\approx 0.632.$

The common misconception is to use $\lambda=3$ without adjusting for the shorter interval. The value of $\lambda$ must always correspond to the interval in the question.

Exam technique

This is HL-only content from AHL 4.17, particularly the skill of choosing an appropriate probability distribution. In an IB response, justify Poisson by explicitly stating “independent events occurring at a uniform average rate,” show any scaling of $\lambda$, then use the GDC and give the final probability to three significant figures unless instructed otherwise.