Page 211 - Probability, Random Variables and Random Processes

CHAP. 53 RANDOM PROCESSES 203

5.58. Consider a Poisson process X(t) with rate A, and suppose that each time an event occurs, it is
classified as either a type 1 or a type 2 event. Suppose further that the event is classified as a type
1 event with probability p and a type 2 event with probability 1 - p. Let X,(t) and X,(t) denote
the number of type 1 and type 2 events, respectively, occurring in (0, t). Show that (X,(t), t 2 0)
and {X,(t), t 2 0) are both Poisson processes with rates Ap and A(1 - p), respectively. Further-
more, the two processes are independent.
We have

First we calculate the joint probability PIXl(t) = k, X2(t) = m].

Note that
P[X,(t) = k, X2(t) = m 1 X(t) = n] = 0 when n # k + m
Thus, using Eq. (5.1 58), we obtain

Now, given that k + m events occurred, since each event has probability p of being a type 1 event and
probability 1 - p of being a type 2 event, it follows that

Thus,

Then

which indicates that X,(t) is a Poisson process with rate Ap. Similarly, we can obtain

and so X2(t) is a Poisson process with rate A(l - p). Finally, from Eqs. (5.170), (5.171), and (5.169), we see
that

Hence, Xl(t) and X2(t) are independent.