The first one should have I think but other than that it seems fine.
If you have problems with latex, post it in the latex subforum, 'tis working fine for me.
Let be independent exponential variables with parameters respectively. Find the density function of .
I get a piecewise density function (for ):
The answer given however is simply that the density function is exponential without further information. I was wondering if my answer is correct?
P.S. I can't seem to do piecewise using normal LaTeX code; is it different in these forums?
How to find the result with little computation (but some nontrivial knowledge): first, the probability that is (classic useful computation); then, by the memoryless property of exponential distribution, conditionally to and to the event , the distribution of is exponential with parameter (note that since this distribution does not depend on , it was unnecessary to condition by ). This gives the " " part: , and symmetry gives the other part.