How do I calculate the UMVUE of $P(X\leq c)$ where $X_i\sim \mathrm{N}(\mu,9)$?
I know that $\mathbf{1}_{\{X_1\leq c\}}$ is an unbiased estimator of $P(X_1\le c)$. Besides, $S=\bar{X}$ is a complete sufficient statistic for $\mu$, so applying Rao-Blackwell once by calculating $\mathrm{E}[\mathbf{1}_{\{X_1\leq c\}}|S=s]$ should give the UMVUE (by Lehmann-Scheffé). How do I proceed from here?