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.