 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-38/2/b/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

The event contribution is an ordinary density integral. Since

f = - F^{'}

and

H^{'} = h

E [H (X) 1_{{T \leq c}}] = \int_{0}^{c} H (t) f (t) d t = - \int_{0}^{c} H (t) d F (t) = - H (c) F (c) + H (0) F (0) + \int_{0}^{c} F (t) h (t) d t = - H (c) F (c) + \int_{0}^{c} f (t) d t .

Consequently

E [H (X) 1_{{T \leq c}}] = 1 - F (c) - F (c) H (c) .

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