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Let us say we have m i.i.d discrete random variables, for the example let us say they are all distributed Binomial with parameter n and p ($x_i \sim Binom(n,p)$).

Now I want to know the chance that k out of m random variables got the exact same value (while the others got different values).

I know how to do this calculation "manually", e.g: grow a tree of all the options, calculate all the conditional probabilities, and get a result.

My question is if there is some "smarter/shorter" (even asymptotic) way of answering such a question.

Thanks.

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When you say you "want to know the chance that $k$ out of $m$ random variables got the exact same value (while the others got different values)", how would you treat cases where $k=2$ and $m=5$ where (a) two of the random variables had the same values, another two had the same values (different from the first pair) and the fifth had a distinct values and (b) two of the random variables had the same values, and the other three had the same values (different from the first pair)? –  Henry Nov 25 '13 at 0:10
    
Good question Henry. How about the maximal number of identically valued observations? –  Tal Galili Nov 25 '13 at 10:33
    
It is easy enough when $k \gt \frac{m}{2}$ since it is $${m \choose k}\sum_j \Pr(X=j)^k \left(1-\Pr(X=j)\right)^{m-k}$$ but much harder otherwise –  Henry Nov 26 '13 at 7:30

1 Answer 1

This is simply the Multinomial distribution, with the PDF of the underlying distribution (Binomial in your case) providing the parameters $Pk$ in the linked definition, and which can then answer every sort of outcome possibility. Not something you'd want to do by hand, but staple in any stats./prob. package like $R$, etc.

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