Distribution of Product of Dependent RV's

1. Aug 18, 2010

PlasticOh-No

Distribution of Product of Dependent RV's

Hello all.
Let's say we have two random variables, say X and Y.
We know the marginal densities for them, say Px(X) and Py(Y).
How do we find the density of Z = X*Y?
The important part here is that X and Y are dependent.
If there are any tips or directions you can point me then great.

2. Aug 19, 2010

SW VandeCarr

An application of the product formula is given here.

If X and Y are dependent then f X,Y (x,y)=f Y|X (y|x) f X(x)=f X|Y (x|y) f Y(y)

Note: E[X,Y]=E[X]E[Y]+Cov [X,Y]

Last edited: Aug 19, 2010
3. Aug 19, 2010

PlasticOh-No

Thanks for the reply. However, note that this paper shows an algorithmic approach to the calculation of the distribution of independent random variables.

From the abstract,
What I need is an understanding of the case when the variables in question are dependent.

4. Aug 19, 2010

PlasticOh-No

Also I am not saying that I need to find the joint density of X and Y.

I need to think about the distribution of Z, when Z = X*Y and all I have to go on are X and Y's marginals.

5. Aug 19, 2010

SW VandeCarr

The Rohatgi integral can handle dependence. That's why I included the formula for f X,Y (x,y) when X and Y are dependent.

Also see:http://en.wikipedia.org/wiki/Product_distribution

Last edited: Aug 19, 2010
6. Aug 19, 2010

PlasticOh-No

I see. So we will be needing the joint distribution.
My mistake, thank you very much for your help.

7. Aug 19, 2010

SW VandeCarr

You're welcome.

8. Aug 20, 2010

SW VandeCarr

Error in post 2: That should be E[XY]=E[X]E[Y]+Cov[X,Y]

Last edited: Aug 20, 2010
9. Oct 15, 2010

PlasticOh-No

Hello again
Can you give tips on also distribution of:
sum or difference on random variables that are
-possibly dependent
-non Gaussian
Thank you

10. Oct 16, 2010

PlasticOh-No

I got it, it is
Z=X+Y
$$f_Z(z)=\int_{-\infty}^{\infty}f(x,z-x)dx [\tex] where f is the joint dist 11. Oct 16, 2010 SW VandeCarr corrected Latex 12. Oct 17, 2010 PlasticOh-No Arrg. Thanks Matey [tex] f_Z(z)=\int_{-\infty}^{\infty}f(x,z-x)dx$$

Shiver me timbers
How does one edit an old post? thanks