Suppose $n$ values, $X_1,...,X_n,$ are generated by a random number generator with normal distribution $N(0,1).$ Suppose that the (sample) mean of $X_1,...,X_n$ is $\mu.$ What is known about the order statistics of $X_1,...,X_n$? (Eg. what is the expected value of $Y_c=\#\{i: X_i>c\}$ as an expression in $n,\mu$ and $c.$ If there is no exact formula, then perhaps value of limit of it as $n\to \infty$?)

Hint: the conditional distribution of $X_i$ given $X_1 + \ldots + X_n = \mu$ is normal ... EDIT: Sorry for just giving a hint before: I was busy. $X_1$ and $S = X_1 + \ldots + X_n$ are jointly normal with means $0$ and covariance matrix $C = \pmatrix{1 & 1 \cr 1 & n\cr}$. So they have joint density $$f_{X_1,S}(x,s) = \dfrac{1}{2\pi \sqrt{n1}} \exp\left(\dfrac{n x^2  x s + s^2}{2(n1)}\right)$$ From that you can compute that the conditional distribution of $X_1$ given $S=s$ is normal with mean $s/n$ and variance $1  1/n$. In particular, $$P(X_1 > c S = n \mu) = 1  \Phi\left(\dfrac{c\mu}{\sqrt{11/n}}\right)$$ where $\Phi$ is the standard normal CDF. The same is true for all other $X_i$, so by linearity of expected value $$E[Y_cS = n \mu] = n \left(1  \Phi\left(\dfrac{c\mu}{\sqrt{11/n}}\right)\right)$$ 

