# pdf of product of two dependent rv

Started by August 24, 2006
```Hello,
I am facing problem in finding the pdf of  Z= AB+CD,
where A, B, C, D are  independent normal random variables  with zero
mean and uneqaul variances.
I could bale to find out the pdf's of each term (i.e. AB or CD).  It is
a well known result that AB has a pdf of   modified Bessel function of
the second kind.
If we add two random variables having the pdf of " modified Bessel
function of the second kind.", assuming both are independent, I am
unable to get the final expression . Since it involves the convolution
of two " modified Bessel function of the second kind.".
Is there any standard expression for finding the pdf of
z= AB+CD?
Best Regards,
-SaiRamesh.

```
```sairamesh@gmail.com wrote:
> Hello,
>                 I am facing problem in finding the pdf of  Z= AB+CD,
> where A, B, C, D are  independent normal random variables  with zero
> mean and uneqaul variances.
> I could bale to find out the pdf's of each term (i.e. AB or CD).  It is
> a well known result that AB has a pdf of   modified Bessel function of
> the second kind.
>  If we add two random variables having the pdf of " modified Bessel
> function of the second kind.", assuming both are independent, I am
> unable to get the final expression . Since it involves the convolution
> of two " modified Bessel function of the second kind.".
> Is there any standard expression for finding the pdf of
> z= AB+CD?
> Best Regards,
> -SaiRamesh.

I think you can numerically obtain the pdf of z (by plotting in matlab)
but I do not think any analytical result can be obtained by convolving
2 Bessel functions

```