# poisson distribution

## 1. binomial distribution

- \(P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\)
- this is the probability of getting \(k\) sucesses in \(n\) time windows

## 2. poisson distribution

- If we let \(n\rightarrow \infty\) and \(p\rightarrow 0\) (I'm omitting a couple things), then we get the poisson distribution
- \(P(X=k)=\frac{\lambda^ke^{-k}}{k!}\)
- that is, as we move from discrete time windows to continuous time, we make our time windows smaller, and have a smaller probability of success in each window
- in the limit, we arrive at the poisson distribution