A complete exponential sum yields the survival UMVUE

Let $n\ge2$ and let $X_1,\ldots,X_n$ be independent exponential random variables with rate $\theta>0$, so that each has density $\theta e^{-\theta x}$ on $(0,\infty)$. Write $T=\sum_{i=1}^n X_i$. Show that the model is a one-parameter exponential family and that $T$ is a complete sufficient statistic. Identify the law of $T$. The target is $\tau(\theta)=P_\theta(X_1>1)=e^{-\theta}$. Prove that $\delta=\mathbf 1\{X_1>1\}$ is unbiased for $\tau(\theta)$. Then compute $$ \delta^*(T)=\mathbb E[\delta\mid T] $$ explicitly, including the case $T\le1$, and verify that the displayed formula does not contain $\theta$. Conclude that $\delta^*$ is the UMVUE of $e^{-\theta}$. Also give the UMVUE of $1/\theta$. Do not replace this calculation by a Poisson zero-count Rao–Blackwellization, and do not treat the problem as a Neyman–Pearson test of a point null.

The joint density is $\theta^n\exp(-\theta T)$ on the positive orthant.