Poisson & normal distributions as approximations for the binomial

[Rasalhague]

These three quotes talk about the use of the Poisson and normal distributions as approximations for the binomial when n is large. The first two quotes here say Poisson is best when p small, and the normal otherwise. The third seems to change the story; it says Poisson is best for large p too. Is there a contradiction here, or is Hoel just nuancing it as he goes along?

It turns out that for very large n there are two well-known density functions that give good approximations to the binomial density function: one when p is very small and the other when this is not the case. The approximation that applies when p is very small is known as the Poisson density function and it defines the Poisson distribution.

- Hoel: Introduction to Mathematical Statistics, 5th ed., p. 64.

In 2.5.1. the Poisson distribution was introduced as an approximation to the binomial distribution when n is large and p is small. It was stated that another distribution gives a good approximation for large n when p is not small. The normal distribution is the distribution with this property.

- Hoel: ibid., p. 81.

The two approximations that have been considered for the binomial distribution, namely the Poisson and normal distributions, are sufficient to permit one to solve all the simpler problems that require the computation of binomial probabilities. In n is small, one uses formula (11) [the binomial density function] directly because the computations are then quite easy. [...] If n is large and p is small or large, the Poisson approximation may be used. In n is large and p is not small or large, the normal approximation may be used. Thus all probabilities have been covered.

- Hoel: ibid., p. 85.

 

[Aimless]

I believe what is intended here is the Poisson distribution works when either p or (1-p) is small, which is what is meant by p small or large. The normal distribution is favored when p~=0.5.

So, "nuance" it is.

 

[pmsrw3]

The Poisson distribution is a good approximation to the binomial when p << 1. If p is large (i.e., 1-p << 1), then N-n (N = # trials, n = # successes) will follow a Poisson distribution. This is what he means by saying the Poisson "may be used" -- not that n will follow a Poisson, but that you can use the Poisson to calculate the distribution of n. The normal distribution is a good approximation when both Np and N(1-p) are large. Hoel doesn't say this, but when all three conditions are met, p << 1, Np >> 1, and N(1-p) >> 1, both the Poisson and the normal are good approximations. I.e., the normal works even for small p or large p as long as N is big enough to compensate.

 

[Rasalhague]

Excellent! Thanks for clearing that up for me.

 

[blue_raver22]

I can confirm that Hoel is not contradicting himself in these quotes. He is simply discussing different scenarios where the Poisson and normal distributions can be used as approximations for the binomial distribution. It is true that the Poisson distribution is best when p is small, but it can also be used for large p. Similarly, the normal distribution is best for large p, but it can also be used for small p. Therefore, there is no contradiction here; Hoel is just highlighting the different scenarios where each approximation can be applied effectively. This shows the versatility and usefulness of these distributions in approximating the binomial distribution for different values of n and p.