Suppose that Y follows an exponential distribution, with mean . Use the method of moment generating functions to show that is a pivotal quantity and has a distribution with 2 df.
Attempt:
so substituting , and I get:
Suppose that Y follows an exponential distribution, with mean . Use the method of moment generating functions to show that is a pivotal quantity and has a distribution with 2 df.
Attempt:
so substituting , and I get:
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's not the case here).
1. is a function of and .
2. The pdf of does not depend on .
Therefore is a pivotal quantity.