
Independence of events
Let X and Y be random variables that both have an exponential distribution with parameters lambda and mu respectively. X and Y are independent.
Define Z = min{X, Y}
(1) Prove that the event {X < Y} is independent of the event {Z > z}
(2) Find E[ e^(z)  X ].

You can calculate the three probabilities.
That's probably not a clever way, but it must work.
$\displaystyle P(X<Y,\min\{X,Y\}>z)=P(X<Y,X>z)$
$\displaystyle ={1\over \mu\lambda}\int_z^{\infty}\int_z^y e^{x/\lambda}e^{y/\mu}dxdy$
Then get
$\displaystyle P(X<Y)={1\over \mu\lambda}\int_0^{\infty}\int_0^y e^{x/\lambda}e^{y/\mu}dxdy$
and
$\displaystyle P(\min\{X,Y\}>z)= P(X>z,Y>z)= P(X>z)P(Y>z)$
$\displaystyle ={1\over \mu\lambda}\int_z^{\infty}e^{x/\lambda}dx\int_z^y e^{y/\mu}dy=e^{z/\lambda}e^{z/\mu}$
Then compare to see if P(AB)=P(A)P(B).