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.