Suppose that $X_1,\ldots,X_n$ are $n$ i.i.d. random variables from the Poisson distribution truncated on the left at $0.$ Find the UMVUE of $P(X_1 =1).$
I am trying to do it by computing the expectation of a function $h(T)$ of my statistic, which is just the sum of $X_i$, and this expectation is equal to $P(X_1 =1),$ so I want to find $h(T),$ which will be the UMVUE, since $T$ is minimal complete sufficient statistic.
The problem is that I obtain a big double sum in this expectation and I don't know how to deal with it.
I have also tried to compute the MLE, but I don't know how to obtain it in this case.
Anyone could help me please?
thanks in advance