McKay's bivariate Gamma distribution

Question

Given the variables $X$ and $Y$, which are correlated, $X\ge0$, $Y\ge0$ and each follow a gamma distribution with different shape parameters, i.e.,$X\sim\Gamma(a_1,\alpha)$ and $Y\sim\Gamma(a_2,\alpha)$. I understood that the Joint PDF $f_{X,Y}(x,y)$ can be obtained by making use of the McKay's bivariate Gamma distribution which applies for the case of different shape parameters.

$\mathbf{First}: $

McKay's PDF has a condition of $Y>X$ (and vice versa), does that mean that in this case there is no situation (in the event space) such that $Y=X$?

$\mathbf{Second}: $

If I want to obtain the following average of another function, denoted by $G(X,Y)$, i.e,

$\mathbb{E}[G(X,Y)]=\int^\infty_0\int^\infty_0 G(X,Y)f_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y$

do I need to average the individual cases when $f_{X,Y}(x,y)$ applies for $X>y$ and $X<y$?

what about the case of $X=Y$?

Answer 1

You can create whole families of joint distributions on $(X,Y)$ such that $X\sim \Gamma(a_1,\alpha)$ and $Y\sim \Gamma(a_2,\beta)$ by using copulas like $$ F_{(X,Y)}(x,y) = \mathbb{P}(X\le x,Y\le y) = \dfrac{F_X(x)F_Y(y)}{1+\varrho (1-F_X(x))(1-F_Y(y))} $$ for $-1\le \varrho \le 1$. The joint distribution is continuous, which means the event $X=Y$ has probability zero.

Now, if you have a specific reason for using McKay's bivariate distribution, with pdf $$ f_{(X,Y)}(x,y) = \alpha^{p+q} x^{p-1} (y-x)^{q-1} \exp\{-\alpha y\} / [\Gamma(p) \Gamma(q)]\,\mathbb{I}_{0\le x\le y} \,, $$ which gives $$ X\sim \Gamma(p,\alpha)\,,\quad Y\sim \Gamma(p+q,\alpha) $$ as marginals, you must compute $\mathbb{E}[G(X,Y)]$ as $$ \int_0^\infty \int_0^y G(x,y)\,\alpha^{p+q} x^{p-1} (y-x)^{q-1} \exp\{-\alpha y\} / [\Gamma(p) \Gamma(q)]\,\text{d}x\,\text{d}y\,. $$

Answer 2

For McKay's distribution $X$ is a Gamma variate that is the sum of a subset of squares taken from the other, $Y$, which is the sum of a larger set of squares. Implying that $Y>X$ with probability $1$. See McKay's original paper:

McKay, A. T. (1934) Sampling from batches. Journal of the Royal Statistical Society—Supplement 1: 207–216.