Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 1 / b

Write $F=(1-ϵ)\Phi +ϵH$ and assume $0<ϵ<1/2$. If $m$ is its median, thenMoreover its density satisfies $f(m)\ge (1-ϵ)\varphi (m)$. Since the standard normal density decreases with $|m|$, the smallest possible density at the median occurs at either endpoint. PutThe two endpoints are $\pm q_{ϵ}$ by normal symmetry. The bound is attained by choosing a contaminating density supported strictly to the right of $q_{ϵ}$, or symmetrically to the left of $-q_{ϵ}$, with zero density at the selected median. Therefore