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Math Help - The difference of exponential random variables

  1. #1
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    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
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  2. #2
    MHF Contributor matheagle's Avatar
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    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.
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  3. #3
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    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.
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  4. #4
    MHF Contributor matheagle's Avatar
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    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.
    Last edited by matheagle; April 15th 2009 at 06:01 PM.
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  5. #5
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    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.
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  6. #6
    MHF Contributor matheagle's Avatar
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    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.
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