Q1: Suppose that is a random sample from a probability density in the (one-parameter) exponential family so that

, where is the indicator function and and do not depend on . Show that is sufficient for .

A1:

= .

If I let , which does not depend on and , where interacts with the data only through . So, by the factorization criterion, is sufficient for .

Q2: Let denote a random sample from the probability density function

for , and otherwise.

a) Show that this density function is in the (one-parameter) exponential family and that is sufficient for .

b) If , show that has an exponential distribution with mean .

Furthermore, in Q2, the book tells us to refrence back to Q1 as a hint.

A2:

I am really stuck on this one. I am not sure how to show something is in the one-parameter exponential family, as I have never seen such a thing before. So, some steps might be nice to see or even an explanation of the process. Any help would be greatly appreciated.