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(iii) N (t) N (s) is Poisson with mean s (r) dr. 0} is a The ﬁrst deﬁnition does not work well in this setting since the interarrival
times ⌧1 , ⌧2 , . . . are no longer exponentially distributed or independent. To
demonstrate the ﬁrst claim, we note that
P (⌧1 > t) = P (N (t) = 0) = e Rt
0 (s) ds 86 CHAPTER 2. POISSON PROCESSES since N (t) is Poisson with mean µ(t) =
P (⌧1 = t) = Rt
P (t1 > t) = (t)e
dt (s) ds. Di↵erentiating gives the
0 (s) ds = (t)e µ(t) Generalizing the last computation shows that the joint distribution
fT1 ,T2 (u, v ) = (u)e
Changing variables, s = u, t = v
f⌧1 ,⌧2 (s, t) = (s)e µ(u) · (v )e (µ(v ) µ(u)) u, the joint density
µ(s) · (s + t)e (µ(s+t) µ(s)) so ⌧1 and ⌧2 are not independent when (s) is not constant. 2.3 Compound Poisson Processes In this section we will embellish our Poisson process by associating an independent and identically distributed (i.i.d.) random variable Yi with each arrival.
By independent we mean that the Yi are independent of each other and of the
Poisson process of arrivals. To explain why we have chosen these assumptions,
we begin with two examples for motivation.
Example 2.1. Consider the McDonald’s restaurant on Route 13 in the southern part of Ithaca. By arguments in the last s...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
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