As $b\to\infty$, the events $\{T_a<T_b\}$ increase to $\{T_a<\infty\}$: every path that reaches $a$ has a finite maximum before doing so. Since $\lambda^b\to0$,
$$
\boxed{\mathbb P(T_a<\infty)=\lambda^{-a}.}
$$
Similarly, as $a\to-\infty$, the events $\{T_b<T_a\}$ increase to $\{T_b<\infty\}$, and
$$
\mathbb P(T_b<T_a)
=\frac{1-\lambda^a}{\lambda^b-\lambda^a}
\longrightarrow1.
$$
Hence
$$
\boxed{\mathbb P(T_b<\infty)=1.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-201/1/c/solution