ppois() — Cumulative Poisson

The chance of q things or fewer happening when you usually expect λ.

ppois(q, lambda, lower.tail = TRUE)

Key idea

• ppois gives the cumulative Poisson chance of AT MOST q events, written P(X <= q) — q events or fewer. (e.g. ppois(4, 5) = 0.440)
• It sums dpois for every count from 0 through q. (e.g. P(0)+...+P(4))
• Use it for q or fewer events given an average rate lambda. (e.g. at most 4 emails)
• For more than q events, take 1 minus the cumulative value. (e.g. 1 - ppois(4,5))
• As q grows large the cumulative chance approaches 1. (e.g. ppois(20,5) = 1)

Definition

$$F(q)=\sum _{k=0}^q\frac{{\lambda }^k{e}^{-\lambda }}{k!}$$

$q$

events counted up to

$\lambda$

the average rate (lambda)

$$P(X\le 3)=e^{-2}\sum _{k=0}^3\frac{2^k}{k!}=0.8571$$

Where you'd use it

• A call centre averages 5 calls an hour. The chance of 4 or fewer this hour is ppois(4, 5) ≈ 44%, so a quiet-ish hour like that happens almost half the time.
• A town averages 1 power cut a month. The chance of at most 2 cuts this month is ppois(2, 1) ≈ 92%, meaning 3 or more would be a notably bad month.
• A busy road averages 8 crashes a year. The chance of more than 10 is the upper tail, ppois(10, 8, lower.tail = FALSE) ≈ 18%, so an unusually crash-heavy year is uncommon.

Frequently asked questions

What does ppois give?

P(X ≤ k), the cumulative Poisson probability. For "more than k" use lower.tail = FALSE.