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Unformatted text preview: , which will be our o cial deﬁnition, is nice because it allows us to
construct the process easily.
Deﬁnition. Let ⌧1 , ⌧2 , . . . be independent exponential( ) random variables.
Let Tn = ⌧1 + · · · + ⌧n for n 1, T0 = 0, and deﬁne N (s) = max{n : Tn s}.
We think of the ⌧n as times between arrivals of customers at a bank, so Tn =
⌧1 + · · · + ⌧n is the arrival time of the nth customer, and N (s) is the number
of arrivals by time s. To check the last interpretation, consider the following
example:
⌧1
0 ⇥
T1 ⌧2 ⇥
T2 ⌧3 ⇥
T3 ⌧4 ⌧5
⇥
T4 s ⇥
T5 Figure 2.1: Poisson process deﬁnitions.
and note that N (s) = 4 when T4 s < T5 , that is, the 4th customer has arrived
by time s but the 5th has not.
Recall that X has a Poisson distribution with mean , or X = Poisson( ),
for short, if
P (X = n) = e n n! for n = 0, 1, 2, . . . 81 2.2. DEFINING THE POISSON PROCESS To explain why N (s) is called the Poisson process rather than the exponential
process, we will compu...
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 Spring '10
 DURRETT
 The Land

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