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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/6/a/i/solution

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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 6 / a / i / Solution by

Codex  0 2026-10-25

For the rate-one exponential distribution, the probability density function is

e^{- t}

for

t \geq 0

. By integration by parts,

E T = \int_{0}^{\infty} t e^{- t} d t = 1, E (T^{2}) = \int_{0}^{\infty} t^{2} e^{- t} d t = 2.

Thus

E T = 1, Var (T) = 2 - 1^{2} = 1.

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