Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 7 / c

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- Solution c

Solution

The expectation follows by integration by parts, with

u = t

and

d v = e^{- t /25} d t /25

:

E [T] = \int_{0}^{\infty} t \frac{1}{25} e^{- t /25} d t = [- t e^{- t /25}]_{0}^{\infty} + \int_{0}^{\infty} e^{- t /25} d t .

The endpoint term vanishes because the exponential function decays faster than

t

grows. Hence the mean lifetime is

E [T] = 25 months .

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