Mean and Variance of the Poisson Distribution

For a Poisson random variable, both the mean and the variance equal the rate parameter $\lambda$ . This lesson computes $E(X)$ , $\text{Var}(X)$ , and $\sigma_X$ for Poisson random variables in concrete scenarios, including ones where the rate must be scaled to match the interval of interest.

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Tutorial

The Mean of a Poisson Random Variable

The mean (or expected value) of a Poisson random variable equals its rate parameter. If $X \sim \text{Poisson}(\lambda)$ , then

E(X) = \lambda.

This matches the interpretation of $\lambda$ itself: it is the average number of events expected in the given interval.

For instance, if a switchboard receives phone calls according to $X \sim \text{Poisson}(12)$ over one hour, then on average we expect

E(X) = 12 \text{ calls per hour}.

The rate parameter scales with the length of the observation interval. If events occur at a constant rate of $r$ per unit time and $X$ counts events over an interval of length $t$ , then $X \sim \text{Poisson}(rt)$ and

E(X) = rt.