# Thread: The difference of exponential random variables

1. ## 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

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.