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I am trying to solve the following problem.

Let there be $n$ urns and let us have $k$ balls. Assume we put every ball into one of the urns with uniform probability. Denote by $X_i$ the random variable counting the number of balls in urn $i$. If $X = min\{X_1,\ldots,X_n\}$, what is $E[X]$?

As a more general question, one could ask: what is the expected value of the minimum of some equally distributed random variables?

I do not see any way of solving it besides using the definition of expected value which results in a nasty expression.

I believe there is some better technique for approaching this kinds of problems.

Anyone happens to know how?

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    Perhaps it does get easier. For $n=2$ and large $k$ it seems that $\dfrac{k}{2} - \sqrt{\dfrac{k}{2\pi}}$ is a reasonable approximation. In general, $\dfrac{k}{n}$ is clearly an upper bound, so perhaps there is a general approximation.2011-05-02

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