 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/14/b/i/solution

For

0 \leq k \leq n

, sum the marginal stopping probabilities over the selected tail:

P (X > x_{k}) = p j = k + 1 \sum n g_{j} + (1 - p) j = k + 1 \sum n d_{j} .

Equivalently it is

p (1 - \sum_{j = 0}^{k} g_{j}) + (1 - p) (1 - \sum_{j = 0}^{k} d_{j})

. At

k = n

, both empty tail sums are zero, as required.

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