Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

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).

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.