I am working on a problem that requires me to find the UMVUE of $Pr[X=0]=e^{-\lambda}$.
There are several issues that I am having so I would really appreciate your help.
1), What does it mean to be an "estimator" of a probability? I understand that estimators can estimate parameters such as the mean or variance of a distribution, but I do not quite get the intuitive meaning of this problem. . .
2), The notes that I am looking at mentions
$$\Bbb{I}_{\{X_1=0\}}$$
is a "natural choice" of an unbiased estimator of $Pr[X=0]$.
I am thinking that because I don't understand intuitively what is going on I am not seeing why an indicator is an estimator.
The notes further proceeds to simplifying
$$E[\Bbb{I}_{\{X_1=0\}}|\sum_{i=1}^n X_i=x]$$
which leads to
$$\left( \frac{n-1}{n} \right)^x$$
I understand the rest of the algebraic portion, but because I don't know why it started off like this it does not stay in my head.
Thank you for your help.