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 $n \to \infty$ and $p \to 0$ (I'm omitting a couple things), then we get the poisson distribution
$P (X = k) = \frac{λ^{k} e^{- k}}{k!}$
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

3. sources

mathoverflow

4. see also

exponential distribution

Created: 2024-07-15 Mon 01:28