I'm trying to solve an exercise in mathematical statistics. The problem is :
If $X_1,...,X_n$ are $i.i.d$ random variables from $N(\mu,1)$, and if $Y_n$ is the maximum order statistics then what is the best unbiased estimator of $P(Y_n >a)$ for real $a$?
I'm trying to use the Lehmann-Scheffe theorem. Since $T=\sum_1^n X_i$ is a complete sufficient statistic for $\mu$ so I want to compute the conditional expectation $E(U|T)$with some nice unbiased estimator $U$.
I tried $U=1$ if $Y_n>a$ and $U=0$ otherwise. But in this case I cannot compute the conditional expectation explicitly. Is my approach valid?
Any hint or answer would be appreciated.