Assume and are independent random variables, both following a exponential distribution. Calculate
The way it's done in the book
(a)
(b)
(c)
I'm wondering is why are they putting the pdf of exponential function (the part in blue)?
in part (b) would is it
in part (c) isn't it , therefore wouldn't it be:
why isn't there a ?
Not quite, since this even can be negative... It should be:
and it is well known that is an exponential random variable of parameter , and it is independent of so, in application of the result of your initial question,
One easily checks that the integral gives the same result. By the way, the computation of the integral gives by induction the general formula: