The Poisson distribution
The Poisson distribution
- $X \sim \text{Po}(\lambda)$ models the number of random events in a fixed interval at a steady rate $\lambda$:
$$P(X = r) = e^{-\lambda}\frac{\lambda^r}{r!}$$
- For a Poisson variable, the mean and variance both equal $\lambda$.
- It approximates the binomial when $n$ is large and $p$ is small.
Practice
For X ~ Po(3), what is the mean of X?
For a Poisson variable, the mean equals λ = 3 (and so does the variance).
Practice
For X ~ Po(3), find P(X = 2) = e⁻³ × 3²/2!. Give your answer to 3 decimal places.
P(X=2) = e⁻³ × 9/2 = 0.0498 × 4.5 ≈ 0.224.
Practice
For X ~ Po(3), what is the variance of X?
For a Poisson variable, the variance also equals λ = 3.
You've got it
Key idea
- Poisson: $P(X = r) = e^{-\lambda}\dfrac{\lambda^r}{r!}$
- mean and variance both equal $\lambda$
- approximates the binomial for large $n$, small $p$