Exponential RV and Conditional Expectation Problem

From: bg (Stridar_at_gmail.com)
Date: 10/14/04 

Date: 14 Oct 2004 03:09:47 -0700

Dear all,

I am working alone through a book (Sheldon Ross's _Probability Models
for Computer Science_). While I have enjoyed the exposition and have
been able to do most problems, I do not understand the math behind one
question. Would anyone be kind enough to show me how to calculate the
conditional expectation in the following problem?

Problem 1.23

Let X,Y be ind. exponential r.v.'s with rates \lambda and \mu, and let
c >= 0.

a) Find E( min(X,Y) | X > c )
b) Find E( min(X,Y) | X > Y + c)

Thank you,
BG

P.S.
For part (a), I have tried breaking the integral into three parts to
remove the min function instead of finding a joint probability
density. Is this a correct method?

For part (b), I am lost. I thought conditioning on Y would work since
it seems min(X,Y) should simply be Y instead of integrating over the
exponential r.v. with parameter \mu+\lambda. However, Ross gives a
later problem to show
    E(min(X,Y) | X > Y + c) = E(min(X,Y)| X>Y) = \frac{1, \lamda +
\mu}
The above technique does not evaluate to this fraction.

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