Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M4 / c

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- Solution c

Solution

The interarrival times are independent and identically distributed with law

Exponential (λ)

. In particular any finite subcollection of the first

r

has joint density

λ^{r} exp (- λ \sum_{j = 1}^{r} x_{j})

on

x_{j} > 0

. Direct integration gives

M_{X_{1}} (θ) = \int_{0}^{\infty} λ e^{- (λ - θ) x} d x = \frac{λ}{λ - θ} (θ < λ) .

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