The Poisson Distribution Explained with Examples

The Poisson distribution gives the probability of a number of events in a fixed interval when they happen independently at a steady average rate $\lambda$. Examples include calls per hour, typos per page or goals per match. The probability of exactly $k$ events is $\frac{e^{-\lambda}\lambda^k}{k!}$.

When to use it

• Events happen one at a time, independently.
• The average rate is constant over the interval.
• There is no fixed upper limit on the count.

The formula

The mean and the variance are both $\lambda$.

Worked example 1: exactly 2

A help desk gets 4 calls an hour on average. The chance of exactly 2 calls in an hour is

Worked example 2: at most 5

With $\lambda = 3$, add $P(X = 0)$ to $P(X = 5)$:

So there is roughly a 92% chance of 5 or fewer events.

Poisson vs binomial

When $n$ is large and $p$ is small, the binomial is well approximated by a Poisson with $\lambda = np$.

Common mistakes

• Using a rate for the wrong interval (convert 4 per hour to 2 per half-hour).
• Forgetting the $k!$ in the denominator.
• Applying it when events are not independent.

Practice questions

1. With $\lambda = 2$, find $P(X = 0)$.
2. With $\lambda = 2$, find $P(X = 1)$.
3. A shop averages 6 customers per hour. What is $\lambda$ for 20 minutes?

Answers: 1) $e^{-2} \approx 0.1353$ 2) $2e^{-2} \approx 0.2707$ 3) 2

Compute any Poisson probability with the Poisson distribution calculator, or compare with the binomial distribution calculator. See also the binomial distribution explained.

Frequently asked questions

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