Poisson distribution

Definition

Let $\lambda > 0$. The Poisson distribution with parameter $\lambda$ is the distribution on $\{0, 1, 2, \dots\}$ with the probability mass function $$p(k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \dots$$ Here $k!$ is the factorial of $k$. One writes $X \sim \operatorname{Poisson}(\lambda)$.

Remarks

The values add up to $1$ because $\sum_{k \ge 0} \lambda^k / k! = e^{\lambda}$. The mean and the variance are both $\lambda$. It is the limit of the binomial distribution with parameters $n$ and $\lambda / n$ as $n \to \infty$. This is the Poisson limit theorem. Siméon Denis Poisson introduced the distribution in 1837.

Explanations

L1 human

The Poisson distribution counts rare events that happen at random, such as calls to a help desk in one hour. If $\lambda = 2$ calls come per hour on average, the chance of no call in an hour is $e^{-2} \approx 0.135$.