Expressing the product and sum of hypergeometric distribution - notation query

Question

I have a population $N$ with $pN$ success states ($p<1$), and I want to answer a question about getting a certain number of successes from several subsequent selections without replacement. If the successive fractions of $N$ are $(v_{x},v_{y},v_{z})$, then I have a formula that yields $(x_{b},y_{b},z_{b})$ and $(x_{t},y_{t},z_{t})$, respectively the lower and upper number of successes of interest per selection. If I want to calculate the probability of getting within these bounds for every selection, I have to sum the probability of every possible choice from these sets, giving a hypergeometric related distribution which looks like this:

$$P_{Total} = \sum_{x = x_{b}}^{x_{t}} \sum_{y = y_{b}}^{y_{t}} \sum_{z = z_{b}}^{z_{t}} \left(\frac{{pN \choose x}{N - pN \choose v_{x}-x}}{N \choose v_{x}}\right) \left(\frac{{pN-x \choose y-x}{N - v_{x} - pN + x \choose v_{y}-y + x}}{N - v_{x} \choose v_{y}}\right) \left(\frac{{pN- y \choose z-y}{N-v_{x} -v_{y} - pN + y \choose v_{z}-z + y}}{N - v_{x} - v_{y} \choose v_{z}}\right). $$

This formula works, but it is a little unwieldy to write down. I have shown it here for 3 cases, but I could like to generalise it for $k$ cases; is there a neater way to write this relationship, perhaps using product operations, $\Pi$ notation or recurrence relations? I've tried a few ways but am not sure if what I've written down is valid, so any advice welcome.

PS: Incidentally, I am pretty sure that even for relatively small values of $k$ the sheer number of permutations gets virtually impossible to account for, but I just want a neat expression for it in principle, even if it's not to be solved directed

Answer 1

We set $x_0=v_0=0$. The general formula with $n$ cases can then be written as \begin{align*} \color{blue}{P_{Total}^{(n)}=\sum_{x_1=x_b^{(1)}}^{x_t^{(1)}}\sum_{x_2=x_b^{(2)}}^{x_t^{(2)}}\cdots\sum_{x_n=x_b^{(n)}}^{x_t^{(n)}} \prod_{q=1}^n\frac{\binom{pN-x_{q-1}}{x_q-x_{q-1}}\binom{N-\sum_{k=0}^qv_k-pN+x_q}{v_q-x_q+x_{q-1}}}{\binom{N-\sum_{k=0}^{q-1}v_k}{v_q}}} \end{align*}

We have $n$ sums with

• indices $x_1,\ldots, x_n$,

• lower limits $x_b^{(j)}, 1\leq j\leq n$ and

• upper limits $x_t^{(j)}, 1\leq j\leq n$

where we sum over a product with $n$ factors.