Past exam of the options I course of the University of Oxford / 2013 / M / 1 / b

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- Solution b

Solution

The first arrival time

T_{1}

, measured in hours after opening, has exponential distribution of rate twelve. Therefore

P (1/4 < T_{1} < 1/2) = e^{- 12/4} - e^{- 12/2} = e^{- 3} - e^{- 6} .

Equivalently, there must be no arrivals in the first quarter hour and at least one in the next; independent increments give

e^{- 3} (1 - e^{- 3})

.

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