# Expected value of sample median given the sample mean

Let $Y$ denote the median and let $\bar{X}$ denote the mean, of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$?

Intuitively, because of the normality assumption, it makes sense to claim that $E(Y|\bar{X}=\bar{x})=\bar{x}$ and indeed that is the correct answer. Can that be shown rigorously though?

My initial thought was to approach this problem using the conditional normal distribution which is generally a known result. The problem there is that since I do not know the expected value and consequently the variance of the median, I would have to compute those using the $k+1$st order statistic. But that is very complicated and I would rather not go there unless I absolutely have to.

Let $X$ denote the original sample and $Z$ the random vector with entries $Z_k=X_k-\bar X$. Then $Z$ is normal centered (but its entries are not independent, as can be seen from the fact that their sum is zero with full probability). As a linear functional of $X$, the vector $(Z,\bar X)$ is normal hence the computation of its covariance matrix suffices to show that $Z$ is independent of $\bar X$.
Turning to $Y$, one sees that $Y=\bar X+T$ where $T$ is the median of $Z$. In particular, $T$ depends on $Z$ only hence $T$ is independent of $\bar X$, and the distribution of $Z$ is symmetric hence $T$ is centered.