# Bivariate distribution: beta and binomial

Consider a pair of RVs $X$ and $Y$, with the following conditional distributions:

$$X | Y=y \sim Binom(L, y)$$

$$Y | X=x \sim Beta(\alpha + x, \nu)$$

where $L$, $\alpha$, and $\nu$; are all positive ($L$ is an integer of course). Is there a name for the joint distribution of $(X,Y)$? Or perhaps for the marginal distribution of $Y$? I think that if $x$ is eliminated from the shape "parameter" of the beta distribution, then $X$ is beta-binomial distributed. But in the above bivariate model, $X=x$ affects the shape parameter for the conditional distribution of $Y$, so I do not think $X$ is beta-binomial distributed. Apologies if the above makes no sense; I am not very knowledgeable about probability and statistics.

• Thanks for the formatting edits, mpiktas. Didn't realize that the CV site engine can process LaTeX-style formatting. Nice. – Stephen Ramsey May 8 '15 at 15:40

\begin{align} P(x,y) &= \frac{\binom{n}{x}}{\beta(\alpha+x,\nu)} \times y^{\alpha + 2x-1 }(1-y)^{n-x+\nu-1} \\&= \frac{\binom{n}{x}}{\beta(\alpha+x,\nu)} \times \frac{y^{\alpha + 2x-1 }(1-y)^{n-x+\nu-1}}{\beta(\alpha+2x,n-x+\nu)} \times \beta(\alpha+2x,n-x+\nu) \end{align}

Integrating w.r.t $y$:

\begin{align}P(x) &= \frac{\binom{n}{x}}{\beta(\alpha+x,\nu)} \times 1 \times \beta(\alpha+2x,n-x+\nu) \\&= \binom{n}{x} \times \frac{\Gamma(\alpha+2x) \Gamma(n-x+v)}{\Gamma(\alpha +n +x + \nu)} \times \frac{\Gamma(\alpha+x+\nu)}{\Gamma(\alpha+x)\Gamma(\nu)} \end{align}

This can be further simplified by using properties of the Gamma function, but doesn't lead to a beta binomial $X$