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There are 3 indistinguishable balls and 3 distinguishable bins.

Let random variable X = #balls in bin 1 and random variable N = #occupied bins

Range(X) = {0,1,2,3} Range(N) = {1,2,3}

A0 = No balls in bin 1, A1 = 1 ball in bin 1, A2 = 2 balls in bin 1, A3 = 3 balls in bin 1,

B1 = 1 occupied bin, B2 = 2 occupied bins, B3 = 3 occupied bins,

How do I find the distribution of each case and eventually the joint distribution?

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  • $\begingroup$ What do you mean by "finding the partition"? $\endgroup$ Apr 5, 2014 at 22:46
  • $\begingroup$ I got confused with some of the definitions. The partition of X would be {A0, A1, A2, A3, A4, A5} and N would be {B1, B2, B3}. $\endgroup$ Apr 5, 2014 at 22:55

1 Answer 1

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For these numbers I'd be tempted to simply temporarily consider the three balls distinguishable, write down all 27 combinations of which balls go in which bins, and count the number of possible combinations for each of the 12 $(X,N)$ outcomes of the joint distribution.

Certainly you can find some symbolic formula for the joint distribution that would be useful if you had more balls or bins, but it would probably involve several binomial coefficients per value -- so in the 3-by-3 case even evaluating the symbolic formula 12 times would probably be more work than enumerating the 27 combinations and counting possibilities for each of the 12 outcomes.

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