Asymptotic Distribution of the Sample Median

Let $X_1, \ldots, X_n$ be i.i.d.\ $\mathrm{Uniform}(0,1)$ and let $M_n$ denote the sample median (the middle order statistic for odd $n$, or the average of the two middle values for even $n$).

The asymptotic theory of order statistics gives $$\sqrt{n}\,(M_n - m) \xrightarrow{d} N\!\left(0, \frac{1}{4[f(m)]^2}\right),$$ where $m$ is the population median and $f$ is the density at $m$.

(a) For $\mathrm{Uniform}(0,1)$, identify $m$ and $f(m)$, and state the asymptotic variance of $\sqrt{n}\,M_n$.

(b) For $n = 400$, approximate $P(M_{400} > 0.54)$.

You may use $\Phi(1.60) \approx 0.9452$.