 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-40/1/b/i/solution

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

Write

X = min (T, c)

and

Δ = 1_{{T \leq c}}

for the right-censored survival time and observed-failure indicator. Since the exponential distribution is continuous, including or excluding equality at

c

makes no difference. Using the tail-integral expression for an expectation,

E [X] = \int_{0}^{c} P (T > u) d u = \int_{0}^{c} e^{- θ u} d u = \frac{1 - e ^{- θ c}}{θ} .

Consequently the expected integrated hazard at the observed endpoint satisfies

E [H (X)] = θE [X] = 1 - e^{- θ c} = P (T \leq c) = E [Δ] .

Administrative censoring therefore stops both the accumulated exposure and the opportunity to record a failure.

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