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Unformatted text preview: Proof. Again we consider only the case k = 2 and check the second deﬁnition
given in Theorem 2.7. It is clear that the sum has independent increments
and N1 (0) + N2 (0) = 0. The fact that the increments have the right Poisson
distribution follows from Theorem 2.4.
We will see in the next chapter that the ideas of compounding and thinning
are very useful in computer simulations of continuous time Markov chains. For
the moment we will illustrate their use in computing the outcome of races
between Poisson processes. 90 CHAPTER 2. POISSON PROCESSES Example 2.5. A Poisson race. Given a Poisson process of red arrivals with
rate and an independent Poisson process of green arrivals with rate µ, what
is the probability that we will get 6 red arrivals before a total of 4 green ones?
Solution. The ﬁrst step is to note that the event in question is equivalent to
having at least 6 red arrivals in the ﬁrst 9. If this happens, then we have at
most 3 green arrivals before the 6th red one. On the other hand if there are 5
or fewer red arrivals in the ﬁrst 9, then we have had at least 4 red arrivals and
at most 5 green.
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
- Spring '10
- The Land