
Propagation of Uncertainty in a Continuous Time Dynamical System 229
0 0.5 1 1.5 2 2.5 3
1
2
3
4
5
6
7
8
t
X(t)
FIGURE 7 .2: A realization of a unit Poisson process.
In addition, the mean function µ of a renewal process satisfies the so-called
renewal equation given by
µ(t) = P
S
(t) +
Z
t
0
p
S
(s)µ(t − s)ds, (7.35)
where P
S
and p
S
respectively denote the cumulative distribution function and
probability density function of the interarrival times of the renewal process.
Interested readers can refer to [92, Chapter 5] for more information on the
renewal process.
To he lp to visualize the Poisson process, a realization of a unit Poisson
process is demonstr ated in Figure 7.2. ...