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 $n\to \infty$?)



Expected number of samples above certain value of a normally distributed variable with a given sample mean
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
$$f\_{X\_1,S}(x,s) = \dfrac{1}{2\pi \sqrt{n-1}} \exp\left(-\dfrac{n x^2 - x s + s^2}{2(n-1)}\right)$$
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,
$$P(X\_1 > c| S = n \mu) = 1 - \Phi\left(\dfrac{c-\mu}{\sqrt{1-1/n}}\right)$$
where $\Phi$ is the standard normal CDF. The same is true for all other $X\_i$,
so by linearity of expected value


$$E[Y\_c|S = n \mu] = n \left(1 - \Phi\left(\dfrac{c-\mu}{\sqrt{1-1/n}}\right)\right)$$


