 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/m/m5/c/solution

Successive interarrival times

W_{1} = T_{1}

and

W_{2} = T_{2} - T_{1}

are independent exponential variables of rate

λ

λ^{2} e^{- λ (w_{1} + w_{2})}

for

w_{1}, w_{2} > 0

. The transformation

t_{1} = w_{1}

t_{2} = w_{1} + w_{2}

f_{T_{1}, T_{2}} (t_{1}, t_{2}) = λ^{2} e^{- λ t_{2}} 1_{{0 < t_{1} < t_{2}}} .

Integrating over

0 < t_{1} < t_{2} < \infty

gives one, as a joint density should.

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