Summing part i and reindexing gives
$$
\boxed{\mathbb P(M_n\geq m)=\mathbb P(X_n=m)+2\sum_{x>m}\mathbb P(X_n=x).}
$$
Subtracting the formula for $m+1$ yields
$$
\boxed{\mathbb P(M_n=m)=\mathbb P(X_n=m)+\mathbb P(X_n=m+1).}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 2 / 12F / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-2/12f/ii/solution