# Fisher's Approximation of a Binomial Distribution

1. Nov 13, 2013

### gajohnson

1. The problem statement, all variables and given/known data

Suppose that X is the number of successes in a Binomial experiment with n trials and
probability of success θ/(1+θ), where 0 ≤ θ < ∞. (a) Find the MLE of θ. (b) Use Fisher’s
Theorem to find the approximate distribution of the MLE when n is large.

2. Relevant equations

Fisher's Approximation in its full form can be see here:

http://grab.by/s1bA

3. The attempt at a solution

The MLE is easy enough to find, but the question I have is one about Fisher's approximation. In the formula for calculating the 1/(tau)^2 portion of the variance, note that you must use the distribution of only one of X1, X2, etc. This makes plenty of sense if you have a collection of some other kinds of i.i.d. variables (normal, exponential, etc.), but what about if you have one binomial experiment made up of n trials?

What does this imply in the binomial case? Do I calculate 1/(tau)^2 using just the entire binomial distribution given in the problem, or do I use something else (one bernoulli trial, maybe)? Thanks!