# Poisson & normal distributions as approximations for the binomial

1. Jul 17, 2011

### 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?

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

- Hoel: ibid., p. 81.

- Hoel: ibid., p. 85.

2. Jul 17, 2011

### 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.

3. Jul 17, 2011

### 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.

4. Jul 17, 2011

### Rasalhague

Excellent! Thanks for clearing that up for me.