Distribution of Product of Dependent RV's

[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.

 

[SW VandeCarr]

Distribution of Product of Dependent RV's

If there are any tips or directions you can point me then great.

An application of the product formula is given here.

http://www.math.wm.edu/~leemis/2003csada.pdf

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]

 

[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,

We present an algorithm for computing the probability density function of the product of two independent random variables

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

 

[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.

 

[SW VandeCarr]

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

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

 

[PlasticOh-No]

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

 

[SW VandeCarr]

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

You're welcome.

 

[SW VandeCarr]

An application of the product formula is given here.

http://www.math.wm.edu/~leemis/2003csada.pdf

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]

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

 

[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

 

[PlasticOh-No]

I got it, it is
Z=X+Y
[tex] f_Z(z)=\int_{-\infty}^{\infty}f(x,z-x)dx [\tex]

where f is the joint dist

 

[SW VandeCarr]

I got it, it is
Z=X+Y
$$f_Z(z)=\int_{-\infty}^{\infty}f(x,z-x)dx$$
where f is the joint dist

corrected Latex

 

[PlasticOh-No]

Arrg. Thanks Matey

$$f_Z(z)=\int_{-\infty}^{\infty}f(x,z-x)dx$$

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