Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Distribution of choice overlap

  1. Aug 12, 2010 #1
    I have N people and each of them has a uniform probability to belong to one of M countries. Now I wonder what is the distribution of the multiplicities?!
    I mean the number of countries with 1 person, 2 persons, 3 persons,... (no matter which country)
    Is there an equation for it?

    And what if the distribution is not uniform, but I know that some countries are more popular?! (a non-uniform distribution?)
    Is there a sensible measure of quantifying this redistribution of popularity?

    And does power law distribution play in at some point?
  2. jcsd
  3. Aug 23, 2010 #2
    If each person has probability p_k of being from country k independently then the distribution of country populations would be multinomial.

    The joint distribution of population multiplicities would be a sum of these multinomial probabilities, which for uniform distribution (p_k=1/M) would simplify using the permutations with repeats formula, to get

    P(N1=n1,N2=n2,...) = (M!/(n1!n2!...))*(1/M)^N*N!/((1!)^n1.(2!)^n2...)

    where N1 is the number of countries with population 1, etc. Not sure how to get the marginal distributions though, or the joint prob for non-uniform.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook