1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Stat mech and binomial distribution

  1. Apr 15, 2014 #1
    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:

    [tex]p_N(N_A) = \frac{N!}{N_A ! (N-N_A)!} p_A^{N_A} p_B^{N-N_A}[/tex]

    Now I'm asked to consider the case where N gets large. Then I need to find [tex]p(N_A;N)[/tex] 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. jcsd
  3. Apr 16, 2014 #2

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    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?
     
  4. Apr 16, 2014 #3
    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Stat mech and binomial distribution
  1. Stat Mech (Replies: 12)

  2. Stat Mech (Replies: 7)

  3. Stat Mech Question (Replies: 0)

Loading...