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Unformatted text preview: iven by
( t)n 1
fTn (t) = e t ·
for t 0
and 0 otherwise. 80 CHAPTER 2. POISSON PROCESSES Proof. The proof is by induction on n. When n = 1, T1 has an exponential( )
distribution. Recalling that the 0th power of any positive number is 1, and by
convention we set 0!=1, the formula reduces to
fT1 (t) = e t and we have shown that our formula is correct for n = 1.
To do the induction step, suppose that the formula is true for n. The sum
Tn+1 = Tn + ⌧n+1 , so breaking things down according to the value of Tn , and
using the independence of Tn and tn+1 , we have
fTn+1 (t) =
fTn (s)ftn+1 (t s) ds
0 Plugging the formula from (2.12) in for the ﬁrst term and the exponential
density in for the second and using the fact that ea eb = ea+b with a =
(t s) gives
Zt n 1
s ( s)
( t s)
ds = e
which completes the proof. 2.2 Deﬁning the Poisson Process In this section we will give two deﬁnitions of the Poisson process with rate
. The ﬁrst...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
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