Let be a random sample from a Gamma distribution with density function
where is a known integer, and is an unknown parameter.
i. Show that the moment generating function of is
ii. Find the constant c such that (that's "c over Y bar", if it does not show up properly on the screen) is an unbiased estimator for , where (X bar) is the sample mean. Calculate the vairance of this estimator and compare it with the Cramer-Rao lower bound.
(This is an exact wording from the book, including X bar which appears out of nowhere)
(i) since Ys are iid.
(ii) I need to find , and I don't think I can simply substitute the mean in the denominator (expected value of the 1 over Y bar) here, so what else can I do?
I could also try . Any pointers?