 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2008/m/3/solution

λ > 0

starts from zero, has independent stationary increments, has unit jumps, and satisfies

N_{t} - N_{s} \sim Poisson (λ (t - s))

for

t > s

. Its inter-arrival times are independent with the exponential distribution of rate

λ

: any finite joint density is

\prod_{i} λ e^{- λ x_{i}}

on positive coordinates.

The second arrival is a sum of two such variables. Convolution gives

f_{T_{2}} (t) = \int_{0}^{t} λ e^{- λ s} λ e^{- λ (t - s)} d s = λ^{2} t e^{- λ t} (t > 0) .

This is a gamma distribution of shape two and rate

λ

. If

S

is the first arrival of the independent rate-

μ

process, exponential independence gives

P (S < T_{1}) = \int_{0}^{\infty} μ e^{- μ s} e^{- λ s} d s = \frac{μ}{λ + μ} .

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