Page 16 - Probability, Random Variables and Random Processes

8 PROBABILITY [CHAP 1

is the conditional probability of an event B given event A. From Eqs. (1.39) and (1.40), we have

P(A n B) = P(A I B)P(B) = P(B I A)P(A) (1 .41 )
Equation (1 .dl) is often quite useful in computing the joint probability of events.
B. Bayes' Rule:

From Eq. (1.41) we can obtain the following Bayes' rule:

1.7 TOTAL PROBABILITY
The events A,, A,, . . . , A, are called mutually exclusive and exhaustive if
n
U Ai = A, u A, u v A, = S and A, n Aj = @ i # j
i= 1
Let B be any event in S. Then

which is known as the total probability of event B (Prob. 1.47). Let A = Ai in Eq. (1.42); then, using
Eq. (1.44), we obtain

Note that the terms on the right-hand side are all conditioned on events Ai, while the term on the left
is conditioned on B. Equation (1.45) is sometimes referred to as Bayes' theorem.

1.8 INDEPENDENT EVENTS

Two events A and B are said to be (statistically) independent if and only if

It follows immediately that if A and B are independent, then by Eqs. (1.39) and (1.40),
P(A I B) = P(A) and P(B I A) = P(B) (1.47)
If two events A and B are independent, then it can be shown that A and B are also independent; that
is (Prob. 1.53),

Then

Thus, if A is independent of B, then the probability of A's occurrence is unchanged by information as
to whether or not B has occurred. Three events A, B, C are said to be independent if and only if

(1 SO)