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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.

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up vote 1 down vote accepted

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).

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Thank you. Suppose the following setting: $X$ and $V$ are uncorrelated standard normals and $Z$ is the square of $X + V$. How does your answer change if the joint PDF of X and Z is needed. The joint PDF should be unique now, no? What about the conditional PDF of $X$ given $Z$. – Iconoclast Oct 20 '13 at 16:33

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