# The difference of exponential random variables

• Apr 14th 2009, 03:28 PM
phillips101
The difference of exponential random variables
Hi

I need to work out the pdf of B-A, where A~exp(alpha) and B~exp(beta). I've tried making a substitution into their joint density of X=A+B, Y=B-A but... It doesn't seem to be working. I get dependencies on whether alpha is greater than beta (which I have no information about) and everything breaks down if alpha=beta.

Can you talk me through/show me how to get the pdf? Thanks :)
• Apr 14th 2009, 08:32 PM
matheagle
You need to make a 2-2 transformation.
you let Y=B-A and it didn't work with X=B+A?
Then maybe letting X=B might be better.
After that you need to integrate out the X rv.
Why don't you post what you've done and I'll look at it.
• Apr 15th 2009, 12:00 AM
phillips101
Ok, trying it with Y=B-A and X=B I get:

Pdf of y to be [(alpha*beta)/(alpha+beta)] *exp(alpha*y)

To get this, I integrated the joint pdf of a,b, which I assume is (alpha*beta)*exp(-alpha*a -beta*b) wrt x after making the substitution. (Jacobian=1). I integrated it between infinite and zero.

Again, I don't think this answer is correct. It doesn't depend on which is bigger, alpha or beta, but... Y should be on the range infinite to minus infinite, and integrating my above pdf on that range doesn't give 1.

Thanks for any help.
• Apr 15th 2009, 11:07 AM
matheagle
You NEED to show your work.
For one I don't know how you're writing your exponential density.
What's important is that when you integrate out the x, you need to integrate x from y to infinity.
• Apr 15th 2009, 01:44 PM
phillips101
Ok, even integrating from y to infinity you get something that doesn't equal 1 when integrated over the whole of R, and so isn't a probability density function.
• Apr 15th 2009, 05:01 PM
matheagle
IF you want me to look over your work, I will need to see it.
And use TeX.
I don't even know how you're defining your exponential density.