dpois() — Exact Poisson probability

The chance of exactly x things happening when you usually expect λ.

dpois(x, lambda)

Key idea

• dpois gives the chance of EXACTLY x rare independent events given an average rate lambda (a Poisson count). (e.g. dpois(4, 5) = 0.175)
• Lambda is the average number of events per interval, like emails per hour. (e.g. lambda = 5)
• Use it when events are counted over time or space with no fixed upper limit. (e.g. calls per minute)
• For a Poisson, both the average and the spread of the counts equal lambda. (e.g. mean = var = 5)
• The chances are highest for x values near lambda. (e.g. peak near 5)

Definition

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

$x$

number of events

$\lambda$

the average rate (lambda)

$e$

about 2.718

$$P(X=3)=\frac{2^3{e}^{-2}}{3!}=\frac{8e^{-2}}{6}=0.1804$$

Where you'd use it

• A shop averages 5 customers an hour. The chance exactly 4 walk in this hour is dpois(4, 5) ≈ 18%, one of the more likely counts since 4 is close to the average.
• A page averages 1 typo. The chance of exactly 2 typos is dpois(2, 1) ≈ 18%, so seeing two slips on a page is not unusual.
• A help desk gets 3 calls a minute on average. The chance of exactly 3 calls in the next minute is dpois(3, 3) ≈ 22% — tied with exactly 2 calls as the most likely outcome.

Frequently asked questions

What is lambda?

The mean number of events in your interval — match it to the interval in the question.

Can k be a decimal?

No. Poisson counts whole events; non-integers give probability 0.