Let $X_1, .., X_n$ be iid $N(\mu, 1)$. We want to estimate $\theta$ for $P(X_i < \lambda)$ for any $\lambda$.
Consider $U$ = \begin{cases} 1 & X_i <= \lambda \\ 0 & X_I > \lambda \\ \end{cases}
$U(x, \theta)$ is inefficient. I want to Rao-Blackwellize $U$ by using $\bar{X}$ as my sufficient statistic. How would I go about doing this?
I think I need to find $E[U \mid \bar{X}$] but I am not sure if this is correct or where to go from here.