Joint Probability Distribution of a Gaussian Random Variable Correlated with a Gamma Random Variable

Question

I want to know if the joint PDF of a Gaussian RV correlated with a Gamma RV can be found in closed form. The correlation is known.

Answer 1

There are lots of ways to construct bivariate distributions from given marginals. One such way is with copulae. Let the continuous random variable $X$ have pdf $f(x)$ and cdf $F(x)$; similarly, let the continuous random variable $Y$ have pdf $g(y)$ and cdf $G(y)$. We wish to create a bivariate distribution $H(x,y)$ from these marginals. The joint distribution function $H(x,y)$ is given by

$$H(x,y) = C(F,G)$$

where $C$ denotes the copula function (to be defined). Then, the joint pdf $h(x,y)$ is given by:

$$h(x,y)=\frac{\partial ^2H(x,y)}{\partial x\partial y}$$

Examples of copula functions are the Morgenstern copula:

$$C = F G ( 1 + \alpha (1-F)(1-G))$$

and the Ali–Mikhail–Haq copula: $$ C = \frac{F G}{1-\alpha (1-F) (1-G))} $$

etc. (where $\alpha$ is a parameter such that $-1 < \alpha < 1$).

The copula scheme you select will determine the correlation. However, please note that there will not be a unique solution … potentially multiple (indeed, infinitely multiple) solutions may exist (whether by copula methodology or other methods).