Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 223 / 1 / a / Solution

Let $m=F^{-1}(1/2)$. Since $F$ has an everywhere positive density, it is continuous and strictly increasing, so $F(m)=1/2$. Membership in the Kolmogorov neighborhood of a distribution givesBy the symmetry of the standard normal distribution,The stated asymptotic-bias formula for the sample median therefore yields