# 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