# Constructing a bivariate distribution from two gamma-distributed random variables with nonlinear dependence?

I've got 2 gamma-distributed random variables $(X,Y)$ with arbitrary scale and shape parameters. Further, $Y$ should be a non-linear function of $X$, lets say $Y=\sqrt{X}$. What I am interested in is the joint probability $F_{X,Y}(\cdot)$.

All suggestions or general comments are welcome.

Thank you in advance

## 1 Answer

OP wrote: I've got 2 gamma-distributed random variables (X,Y) with ... say $Y=\sqrt{X}$.

Your question is internally inconsistent. In particular, if $X$~Gamma$(a,b)$ with pdf $f(x)$, say:

$$f(x) =\frac{x^{a-1} e^{-\frac{x}{b}}}{b^a \Gamma (a)}, \text{ for } x > 0$$

... and $Y =\sqrt{X}$, then the pdf of $Y$, say $g(y)$, is:

$$g(y) = \frac{2 b^{-a} y^{2 a-1} e^{-\frac{y^2}{b}}}{\Gamma (a)}, \text{ for } y > 0$$

... which is not Gamma$(\alpha, \beta)$, as originally assumed.

• Ok, my formulation was awkward. What I wanted to say is, that the dependence is nonlinear and we could assume something like $E(x_i|Y) = \sqrt{x_i}$. I hope its clear now?! – PeGre Oct 15 '13 at 13:18