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Unformatted text preview: j ) (4.9) 4.2. COMPUTING THE TRANSITION PROBABILITY 125 Introducing matrix notation again, we can write
p0 = pt Q
t (4.10) Comparing (4.10) with (4.7) we see that pt Q = Qpt and that the two forms
of Kolmogorov’s di↵erential equations correspond to writing the rate matrix on
the left or the right. While we are on the subject of the choices, we should
remember that in general for matrices AB 6= BA, so it is somewhat remarkable
that pt Q = Qpt . The key to the fact that these matrices commute is that
pt = eQt is made up of powers of Q:
Q · eQt = 1
X n=0 Q· 1
X (Qt)n
(Qt)n
=
· Q = eQt · Q
n!
n!
n=0 To illustrate the use of Kolmogorov’s equations we will now consider some
examples. The simplest possible is
Example 4.7. Poisson process. Let X (t) be the number of arrivals up to
time t in a Poisson process with rate . In order to go from i arrivals at time
s to j arrivals at time t + s we must have j i and have exactly j i arrivals
in t units of time, so
( t)j i
pt (i, j ) = e t
(4.11)
(j i)!
To check the di↵erential equation,...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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