# Urn problem (indisting. objects into distinguishable urns)

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1. Oct 7, 2016

### fignewtons

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. Oct 7, 2016

### Ray Vickson

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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