Suppose $c>1$ and $\mathbb E[T_c]<\infty$. Part (a) would give
$$
\mathbb E[S_{T_c}^2]=\mathbb E[T_c].
$$
But $T_c\geq1$ and its defining strict inequality gives $S_{T_c}^2>c^2T_c$ almost surely. Taking expectations would yield
$$
\mathbb E[T_c]>c^2\mathbb E[T_c],
$$
which is impossible because $c^2>1$. Part (b) shows that $T_c$ is nevertheless finite almost surely. Consequently
$$
\mathbb E[T_c]=\infty.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-201/1/c/solution