Let . Then using the above theorem, .
It is well known (and easy to prove) that if follows an exponential distribution with mean then .
This is readily recognised as the moment generating function for an an exponential distribution with mean equal to 2.
Therefore the pdf of is .
(I'm not sure where the two degrees of freedom comes from because that's not the case here).
1. is a function of and .
2. The pdf of does not depend on .
Therefore is a pivotal quantity.