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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
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A balls and urns model for a hashing problem
I think I missed something in the model. It seems to me that $r$ is uniform in $\{1,\dots,N\}$.
$$
\mathbb{P}(r=x)=\frac{N-1}{N}\times \frac{N-2}{N-1} \times \dots \frac{N-(x-1)}{N-x+2}\times \frac{1 …