A problem asks that based on sample of size n from a Normal($\theta$,$1$) distribution, we find the Uniformly Minimum Variance Unbiased Estimator (UMVUE) for:  $$p=\Phi(x-\theta)$$  Where $\Phi$ is the CDF of the normal distribution.  My problem is understanding how can I find an estimator for something that is not a parameter of the distribution?  I have tried guessing an estimator and improving it, but again, get stuck with estmating a function instead of a parameter.
Suppose that $X_1,X_2,...,X_n$ is a random sample from a normal distribution, $X_i\sim N(\mu,9)$.  Find the UMVUE (uniformly minimum variance unbiased estimator) of $P(X\le c)$ where $c$ is a known constant.  Do this by finding the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$ and apply the Rao-Blackwell theorem with $T=u(X_1)$, where $u(x_1)=1$ if $x\le c$ and zero otherwise.  So clearly the parameter that we're trying to estimate given the problem description can be written as $\Phi\left( \frac{c-\mu}{3} \right)$ where $\Phi$ represents the CDF of the standard normal distribution.  However, I get stuck when I need to try and find the conditional distribution.  Without knowing the joint distribution or the other conditional distribution, how can I possibly find the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$?
I have often heard people say "x of his" or "x of mine". But since his and mine point out ownership, isn't using "of" here doubles that ownership? Wouldn't it be more appropriate to say "x of him" or "x of them" etc.?  Example:     I was a good friend of your dad's. He told me all about you` SEO (or other) problems with a 404? How do I explain current and voltage to a lay man? A man asked me this question in suburban train when I told him that I am an electrical engineer. Of course I replied him that current is the flow of electrons and in order for the current to flow there should be a potential difference between them. But the man didn't know what electron or potential difference was either. How are character special files and block special files different from regular files in a Unix-like system? Why are they called “character special” and “block special” respectively? Why Can't I duck reverb on the mix bus in Unity? i'm surprised there is so little documentation on this issue. 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Awk: if pattern not found add some text at the end of file Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$. Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G   has subgroup of order $\operatorname{lcm}(n,m)$.   This is a statement that my lecturer mentioned in my (beginners') Abstract Algebra class. I'm not sure I understand why it's true.  What I have so far: Use the abelian group structure theorem on $\langle H, K\rangle$ (finite group generated by $H$ and $K$). Then $\langle H,K\rangle=C_{a_1}C_{a_2}\dotsm$ and $n, m |\langle H,K\rangle$. Which means it's also true that $\operatorname{lcm}(n,m)|\langle H,K\rangle$. Can I leverage this to say there's a subgroup of order $\operatorname{lcm}(n,m)$? 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