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IMM - DTU 02407 Stochastic Processes2001-10-1BFN/bfn

Solution for exercise 6.8.1 in Grimmett and Stirzaker

Intensity The probability of having no flying objects in ∆t is

(1− λ∆t+ o(∆t))(1− µ∆t+ o(∆t)) = 1− (λ+ µ)∆t+ o(∆t)

Correspondingly we find that the probability of having one is

(λ+ µ)∆t+ o(∆t)

Intervals Let Ti by the time to the next event of type i = 1, 2 (fly,wasp). Then the timefor the next flying object is

T = min{T1, T2} P{T ≤ t} = 1−P{T1 > t∩T2 > t} = 1−e−λte−µt = 1−e−(λ+µ)t