Applying part i to the convex function $-g$ gives
$$
\mathbb P(W-\mathbb EW\leq-t)
=\mathbb P((-W)-\mathbb E(-W)\geq t)
\leq e^{-t^2/2}.
$$
This proves probability (iv). Thus both requested tails of the concave function have the claimed bound.


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 4 / c / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-208/4/c/iv/solution