I am trying to calculate the UMVUE for $Pr[X_1\le 2]$ where
$X_1,...X_n \sim_{iid} Exp(\theta) $.
I know that the average number of incidents where the random variable is less than or equal to $2$ is an unbiased estimator of $Pr[X_1\le2]$. Thus,
$$E \left[\frac{\sum_{i=1}^n\Bbb{I}_{X_i \le 2}}{n} \right] = Pr[X_1\le2]$$
The problem that I have is here.
Utilizing Lehmann-Scheffe's Theorem I want to evaluate
$$E \left[\frac{\sum_{i=1}^n\Bbb{I}_{X_i \le 2}}{n} | \sum_{i=1}^nX_i \right] $$
but I am not sure how to procede from here.
Letting $Y=\sum_{i=1}^nX_i$ I know that $Y \sim \Gamma(n,\theta)$ so I know its distribution and I have
$$nE \left[\sum_{i=1}^n \Bbb X_i\le 2 | Y \right]$$