 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/4/b/iii/solution

Only upward and downward moves change the value, so the conditional expectation is

E [R_{k + 1} ∣ R_{k} = r] = r + \frac{( n - r ) ^{2} - r ^{2}}{n ^{2}} = 1 + (1 - \frac{2}{n}) r .

Thus

α = 1, β = 1 - 2/ n

 New to topics? Read the docs here!