Theorem(easy to prove):The moment generating function of is where is the moment generating function of Y.

Let . Then using the above theorem, .

It is well known (and easy to prove) that if follows an exponential distribution with mean then .

Therefore .

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'snotthe case here).

1. is a function of and .

2. The pdf of does not depend on .

Therefore is a pivotal quantity.