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obst12
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Homework Statement
Let X and Y be independent exponential random variables with parameters a and b. Calculate E(X|X+Y).
Homework Equations
The Attempt at a Solution
I'm pretty sure I have it, just want to make sure.
Joint density for X and Y is abe^(-ax)e^(-by) for x,y>0. Let Z=X and W=X+Y so X=Z and Y=W-Z. The Jacobian is 1, so the joint density is
abe^(-az)e^(-bw+bz)=abe^(+bz-az) e^(-bw)
Marginal density for w is
\int_0^ w abe^(bz-az) e^(-bw) dz=ab(e^(bw-aw)-1) /(b-a)
So then conditional density is
(b-a)e^(bz-az) e^(-bw)/(e^(bw-aw)-1)
Then just integrate z from 0 to infinity to get
e^(-bw)/(1-e^(bw-aw))
Is this okay?