# Poisson Distribution $ P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}, \quad 0, 1, 2, \dots $ This holds according to [[taylor-series|Taylor series]]. $ E(X) = \text{Var}(X) = \lambda $ > The Poisson distribution is often used in situations where we are counting the > number of successes in a particular region or interval of time, and there are > a _large number_ of trials, each with a _small probability_ of success. $\lambda$ is the _rate_ of the rare events. The sum of independent Poissons is Poisson -- proved by [[lotp|LOTP]]. ## Poisson Approximation Poisson approximation for small $p_j$s, aka the _law of rare events_. $ X = \sum_{j=1}^n I(A_j) $ Poisson approximation for [[birthday-problem|Birthday Problem]]. $ \lambda = \binom m2 \frac1{365} $