Given an observation $X \sim N(\mu, 1)$, I have to find the MLE of $P(|X|>1)$.
I know that the MLE of $\mu$ is just $X$, since we have only one observation. I also computed $$P(|X|>1) = 1-P(-1<X<1) = 1- P(-1-\mu< X-\mu< 1-\mu)=1 -[ \Phi(1-\mu)- \Phi(-1-\mu)],$$ where $\Phi(x)$ is the CDF of a Standard Normal variable. Then, we can state that, by the invariance property of MLE, the MLE of $P(|X|>1)$ is $1 -[ \Phi(1-\hat{\mu}_{MLE})- \Phi(-1-\hat{\mu}_{MLE})] = 1- \Phi(1-X) + \Phi(-1-X)$.
I need to compare this MLE to the UMVUE of $P(|X|>1)$, so I am looking for an explicit form of the MLE. But I am not sure how to get it from what I got.