Q: If denote a random sample from a geometric distribution with paramter , show that is sufficient for .

A:

.

So, I have that and . Since the unkown parameter only interacts with the data through , is sufficient for .

Now, can I just rewrite the sum as follows to show that is sufficient for ,

?

Also, would it be alright if I started the index from zero or do I have to start from one, since the collection of random variables in the problem statement are started at ? I think that would help clean things up a bit.

Thanks