Example
Example 5.13 Suppose that people immigrate into a territory at a Poisson rate λ = 1 per day. (a) What is the expected time until the tenth immigrant arrives? (b) What is the probability that the elapsed time between the tenth and the eleventh arrival exceeds two days?
Solution: (a) E[S10 ] = 10/λ = 10 days. (b) P{T11 > 2} = e−2λ = e−2 ≈ 0.133.
Proposition 5.1 also gives us another way of defining a Poisson process. Suppose we start with a sequence {Tn , n ⩾ 1} of independent identically distributed exponential random variables each having mean 1/λ. Now let us define a counting process by saying that the nth event of this process occurs at time Sn ≡ T1 + T2 + · · · + Tn The resultant counting process {N(t), t ⩾ 0}∗ will be Poisson with rate λ. ∗
A formal definition of N(t) is given by N(t) ≡ max{n: Sn ⩽ t} where S0 ≡ 0.
5.3 The Poisson Process
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