Binomial Approximation

A Bin(n,p) distribution is the sum of n Bernoulli variables, so the Central Limit Theorem says that it will approximate to a Normal distribution if n is large enough. n could be a few dozen, less for a symmetrical distribution and more for a very non-symmetrical distribution.


For the Poisson distribution E[X] = var[X] = μ. A Bin(n,p) distribution has E[X] = np and var[X] = npq. So for a Binomial distribution to approximate to a Poisson distribution np ≈ npq, i.e. q ≈ 1. For example, if n = 1000 and p = 0.01, then E[X] = 10 and var[X] = 9.9, and a Poi(10) distribution could be considered as an approximation, depending on any other relevant criteria.


The Bin(n,p) distribution has an upper limit of n, while neither the Normal nor Poisson distributions have an upper limit.