Impulses arrive at a circuit according to a Poisson Process at an average rate of λ per minute. This rate is not observable, but the numbers X1, ..., Xn of impulses that arrived during n successive one-minute periods are observed. It is desired to estimate the probability e−λ that the next one-minute period passes with no impulsess.
An extremely crude estimator of the desired probability is
i.e., it estimates this probability to be 1 if no impulses arrived in the first minute and zero otherwise.
The sum

I am confused on how to show that this would be a suffcient statistic. Also I know that
Is the Rao-Blackwell estimator but I cannot manipulate the conditional distribution, we have not reached that point in our probability theory class.

I believe it should become the below statement; however, I am unable to figure out how to fill in the details.
Lastly, I don't understand how one would show that Sn is complete, and thus prove that this estimator is the UMVUE. I sincerely appreciate any help in understanding these concepts.