I'm interested to find out what is the resulting distribution for a discrete random variable $Y$ whose probability mass function is defined as:
$$ Pr(Y = k) = C \cdot \frac{Pr(X_1 = k) \cdot Pr(X_2 = k)}{Pr(X_3 = k)} $$ for $k = 0, 1, 2,\ldots$, where each of the (independent) $X_i$'s follows a Negative Binomial distribution with $X_i \sim NB(r_i, p_i)$ for $i = 1, 2, 3$, while $C$ is the normalizing constant.
I've managed to show that when $X_i \sim Poisson(λ_i)$ then $Y \sim Poisson(\frac{λ_1 λ_2}{λ_3})$ but have so far failed to get somewhere with the negative binomial case.
Starting from:
$$ Pr(X_i = k) = {k + r_i - 1 \choose k} {p_i}^{r_i} (1 - p_i)^k $$
I've managed to find (I think!) that the pmf of $Y$ would include this quantity: $$ \frac{{p_1}^{r_1} {p_2}^{r_2}}{{p_3}^{r_3}} \cdot \bigg[\frac{(1-p_1)(1-p_2)}{(1-p_3)}\bigg]^k $$ and other factorials / Gamma functions involving $r_1, r_2, r_3$ and $k$, but couldn't reach the end. My gut feeling (backed by some simulations) is that $Y$ will also follow a negative binomial distribution but can't prove if that's the case and, if so, what the corresponding expressions for $r_y$ and $p_y$ (in terms of the $r_i$'s and the $p_i$'s) will be.
Any help will be much appreciated. Thanks!