$k$ identical balls are thrown in $n$ urns at random. There may be more than one ball per urn. Let the random variable $W$ be the number of urns that are not empty. What would be the expression of the cumulative distribution function $P(W\le x)$? I am mostly looking for an approximation, because exact results are not required and it's faster to compute.
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My ultimate objective is to generate a random number that follows $W$. I started by finding the probability distribution function of $W$:
$$ P(W=x) = \frac{\frac{n!}{(n-x)!}{k \brace x}}{n^k} $$
where ${k \brace x}$ denotes the Stirling numbers of the second kind.
Then, I would evaluate $P(W\le x)=\sum_{i=1}^{x}{P(W=i)}$, but that becomes inpractical as $n$, $k$ or $x$ get big. Computing $P(W\le x)$ is done in $\mathcal{O}\left(n\right)$, which is no faster than generating $n$ random events and counting the non-empty urns, hence the need for a faster approach.