7. Poissonization

A binomial \((n, p)\) random variable has a finite number of values: it can only be between 0 and \(n\). But now that we are studying the behavior of binomial probabilities as \(n\) gets large, it is time to move from finite outcome spaces to spaces that are infinite.

Our first example of a probability distribution on infinitely many values is motivated by the approximation we have developed for the binomial \((n, p)\) distribution when \(n\) is large and \(p\) is small. This is the family of Poisson distributions, which has powerful relations with many other distribution families.