Page 178 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 5) RANDOM PROCESSES

DEFINITION 5.6.1
A counting process X(t) is said to be a Poisson process with rate (or intensity) 1(> 0) if

X(t) has independent increments.
The number of events in any interval of length t is Poisson distributed with mean At; that is, for
all s, t > 0,

It follows from condition 3 of Def. 5.6.1 that a Poisson process has stationary increments and that
E[X(t)] = At
Then by Eq. (2.43) (Sec. 2.7C), we have
Var[X(t)] = At
Thus, the expected number of events in the unit interval (0, I), or any other interval of unit length, is
just A (hence the name of the rate or intensity).
An alternative definition of a Poisson process is given as follows :

DEFINITION 5.6.2
A counting process X(t) is said to be a Poisson process with rate (or intensity) A(>O) if
1. X(0) = 0.
2. X(t) has independent and stationary increments.
3. P[X(t + At) - X(t) = 11 = A At + o(At)
4. P[X(t + At) - X(t) 2 21 = o(At)
where o(At) is a function of At which goes to zero faster than does At; that is,

om)
lim - - - 0
at-o At
Note : Since addition or multiplication by a scalar does not change the property of approaching zero,
even when divided by At, o(At) satisfies useful identities such as o(At) + o(At) = o(At) and
ao(At) = o(At) for all constant a.
It can be shown that Def. 5.6.1 and Def. 5.6.2 are equivalent (Prob. 5.49). Note that from condi-
tions 3 and 4 of Def. 5.6.2, we have (Prob. 5.50)
P[X(t + At) - X(t) = 0] = 1 - 1 At + o(At) (5.59)
Equation (5.59) states that the probability that no event occurs in any short interval approaches unity
as the duration of the interval approaches zero. It can be shown that in the Poisson process, the
intervals between successive events are independent and identically distributed exponential r.v.'s
(Prob. 5.53). Thus, we also identify the Poisson process as a renewal process with exponentially
distributed intervals.
The autocorrelation function Rx(t, s) and the autocovariance function Kdt, s) of a Poisson
process X(t) with rate 1 are given by (Prob. 5.52)