# Stat mech and binomial distribution

1. Apr 15, 2014

### erogard

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

Suppose that particles of two different species, A and B, can be chosen with
probability p_A and p_B, respectively.

What would be the probability p(N_A;N) that N_A out of N particles are of type A?

3. The attempt at a solution

I figured this would correspond to a binomial distrib:

$$p_N(N_A) = \frac{N!}{N_A ! (N-N_A)!} p_A^{N_A} p_B^{N-N_A}$$

Now I'm asked to consider the case where N gets large. Then I need to find $$p(N_A;N)$$ using 1) Stirling appox. and 2) the central limit theorem.

Not sure how to approach 1) since there are no log in my expression. Actually I don't quite get how to do 2) either. Any help would be much appreciated.

2. Apr 16, 2014

### Matterwave

1) You can introduce the logs! And then remove the logs by exponentiation after you've made the approximation.

2) You should just apply the central limit theorem directly. Do you recall what the CLT says exactly?

3. Apr 16, 2014

### erogard

So I did 1) by taking the log and exponentiating to simplify.

Regarding 2), I'm looking at http://en.wikipedia.org/wiki/Central_limit_theorem and I'm assuming the Linderberg version is the relevant one here. Not too sure how to apply it, however. Do you know of any simpler formulation of the theorem that would apply to my case? My book does not state it formally. Thanks!