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Urn problem (indisting. objects into distinguishable urns)

  1. Oct 7, 2016 #1
    1. The problem statement, all variables and given/known data
    I have n balls and m urns numbered 1 to m. Each ball is placed randomly and independently into one of the urns.
    Let Xi be the number of balls in urn number i.
    So X1+....+Xm = n
    What is the distribution of each Xi?
    What is EXi and VarXi
    What is E[XiXj] given i≠j
    What is Cov(X1,Xj?

    2. Relevant equations
    Cov(XY)=Exy(XY)-Ex(X)Ey(Y)

    3. The attempt at a solution
    I read: https://www.artofproblemsolving.com/wiki/index.php?title=Distinguishability and identified this as the last case.
    I understand that there (n+m-1)ℂ(m-1) ways to place the balls but not how to describe this in the form of a pdf so that I can find expectation and variance and such.
     
  2. jcsd
  3. Oct 7, 2016 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Clearly, for a single urn, the distribution of the number ##X_i## in urn i is the same for any i = 1,2, ...,n, so we might as well look at urn 1. For urns i and j ≠ i the bivariate distribution of ##(X_i,X_j)## is the same for any pair i and j, so we might as well look at urns 1 and 2.

    To find the marginal distribution of ##X_1##, just look at it as a problem having two urns: 1 and not-1. For each object, the probability it goes into urn 1 is 1/m, while the probability it goes into urn not-1 is (m-1)/m.

    For urns 1 and 2 look at it as a three-urn problem with urns 1, 2 and not-12. For each object, p(1) = p(2) = 1/m and p(not-12) = (m-2)/m.
     
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