0
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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?! $\endgroup$ – PeGre Oct 15 '13 at 13:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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