I have a situation where I will draw a random number of balls from an urn with
- $r$ red balls
- and $b$ blue balls,
with $N=r+b$. The number drawn is $k$, and I know the distribution $k$ comes from.
I would like a clean formula for the probability of $u$ distinct red balls. Ideally in some form of generating function.
Thus far I've found that the probability of getting $\ell$ red balls (not necessarily distinct) is
$$\left(\frac{r}{N}\right)^\ell \frac{1}{\ell!} \mu^{(\ell)}(b/N)$$
where $\mu(x)$ is the probability generating function of $k$ and $\mu^{(\ell)}$ denotes the $\ell$-th derivative of $\mu$.
So I can get to a sum over all $\ell$ of the probability of $\ell$ red balls times the probability of $u$ unique balls among the $\ell$ red balls. I'd like a compact way to express this though. I haven't been able to come up with it.
aside
One nice property of the expression for the probability of $\ell$ red balls is that if we replace $r/N$ with $x-b/N$ then this is the $\ell$-th term of the Taylor Series expansion for $\mu(x)$ centered at $b/N$. And in particular if $x=1$ then $1-b/N = r/N$. So summing these terms over all $\ell$ will give $\mu(1)$, which is $1$.