Page 209 - Probability, Random Variables and Random Processes

CHAP. 51 RANDOM PROCESSES

From Eqs. (5.56) and (5.57),
E [X(t)] = It Var [X(t)] = It
Now, the Poisson process X(t) is a random process with stationary independent increments and X(0) = 0.
Thus, by Eq. (5.103) (Prob. 5.23), we obtain
Kx(t, s) = o12 min(t, s) = I min(t, s) (5.1 62)
since a12 = Var[X(l)] = I. Next, since E[X(t)]E[X(s)] = A2ts, by Eq. (5.10), we obtain
Rx(t, s) = I min(t, s) + 12ts (5.1 63)

Show that the time intervals between successive events (or interarrival times) in a Poisson
process X(t) with rate 1 are independent and identically distributed exponential r.v.'s with
parameter A.
Let Z,, Z,, . . . be the r.v.'s representing the lengths of interarrival times in the Poisson process X(t).
First, notice that {Z, > t) takes place if and only if no event of the Poisson process occur in the interval
(0, t), and thus by Eq. (5.154),

Hence Z, is an exponential r.v. with parameter I [Eq. (2.49)]. Let f,(t) be the pdf of Z,. Then we have

which indicates that Z, is also an exponential r.v. with parameter I and is independent of Z,. Repeating the
same argument, we conclude that Z,, Z,, . . . are iid exponential r.v.'s with parameter I.

Let T,, denote the time of the nth event of a Poisson process X(t) with rate A. Show that T, is a
gamma r.v. with parameters (n, 1).
Clearly,

where Z,, n = 1, 2, . . . , are the interarrival times defined by Eq. (5.149). From Prob. 5.53, we know that Z,
are iid exponential r.v.'s with parameter I. Now, using the result of Prob. 4.33, we see that T, is a gamma
r.v. with parameters (n, A), and its pdf is given by [Eq. (2.76)] :

The random process {T,, n 2 1) is often called an arrival process.

Suppose t is not a point at which an event occurs in a Poisson process X(t) with rate A. Let W(t)
be the r.v. representing the time until the next occurrence of an event. Show that the distribution
of W(t) is independent of t and W(t) is an exponential r.v. with parameter A.
Let s (0 2 s < t) be the point at which the last event [say the (n - 1)st event] occurred (Fig. 5-14). The
event (W(t) > 2) is equivalent to the event