Find PDF

**S_David** · Jun7-09, 01:03 PM

Hello,

Suppose that:

where

for i=1 and 2 are independent and identically distribuited exponential RVs.

can we find the PDF of Z?

Regards

**EnumaElish** · Jun7-09, 03:12 PM

You should note that the event {Z = z} is equivalent to {X1 + X2 + X1 X2 = z} or {X1 = (z - X2)/(1 + X2)}. You can use this and use convolutions (example).

**S_David** · Jun7-09, 05:18 PM

Originally Posted by EnumaElish

You should note that the event {Z = z} is equivalent to {X1 + X2 + X1 X2 = z} or {X1 = (z - X2)/(1 + X2)}. You can use this and use convolutions (example).

Right, But I want to find the PDF directly, not from differentiating the CDF, if possible. Because these RVs are, actually, not exponentials, but I said so to simplify the problem statement. So I want to avoid the derivative operation, which complicates the whole stituation.

I say the following:

let

and

, then

. But we need to evaluate joint PDF of W and Y. Is this approach in the right way?

**SW VandeCarr** · Jun8-09, 12:49 AM

Originally Posted by saeddawoud

Hello,

Suppose that:

where for i=1 and 2 are independent and identically distribuited exponential RVs.

can we find the PDF of Z?

Regards

You may want to check the f distribution. The PDF is a bit complicated and I don't have Latex, but you can look it up.

**EnumaElish** · Jun8-09, 01:21 AM

You can derive the pdf directly through convolution.

**S_David** · Jun8-09, 07:28 AM

Originally Posted by EnumaElish

You can derive the pdf directly through convolution.

If we assume that

where

and

, then we need to find the joint PDF $LaTeX Code: f_{W,Z}(w,z)$ , which can be found using Jacobian transformation.

If we proceed using this, we have:

$LaTeX Code: X_1=T_1^{-1}=\\frac{W+Z-X_2}{1+X_2}$ and $LaTeX Code: X_2=T_2^{-1}=\\frac{W+Z-X_1}{1+X_1}$

Then

$LaTeX Code: F_{W,Z}(w,z)=f_{X_1,X_2}(x_1=T_1^{-1},x_2=T_2^{-1})|J|$

where

is the magnitude of the Jacobian which will be zero in this case!!!!

Is here anything wrong I did?

Regards

**EnumaElish** · Jun8-09, 10:48 AM

You can write X1 as (z - X2)/(1 + X2). Then study the wiki example with normal distribution. How is that example similar to your problem?