Page 26 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 11 PROBABILITY

since A, n Aj = @ for i # j. Thus, by axiom 3, we have

which indicates that Eq. (1.31) is also true for n = k + 1. By axiom 3, Eq. (1.31) is true for n = 2. Thus, it is
true for n 2 2,.

1.28. A sequence of events {A,, n 2 1) is said to be an increasing sequence if [Fig. 1-10(a)]
A, c A2 c c A, c Ak+l c
whereas it is said to be a decreasing sequence if [Fig. 1-10(b)]

If (A,, n 2 1) is an increasing sequence of events, we define a new event A, by
CC,
A, = lim A, = U A,
n+co i= 1
Similarly, if (A,, n 2 1) is a decreasing sequence of events, we define a new event A, by
02
A, = lim A, = r)
n+w i= 1
Show that if' {An, n 2 1) is either an increasing or a decreasing sequence of events, then
lim P(A,) = P(A ,)
n-rn
which is known as the continuity theorem of probability.
If (A,, n 2 1) is an increasing sequence of events, then by definition

Now, we define the events B,, n 2 1, by

Thus, B, consists of those elements in A, that are not in any of the earlier A,, k < n. From the Venn
diagram shown in Fig. 1-11, it is seen that B, are mutually exclusive events such that
n n a, 00
U Bi = U A, for all n 2 1, and U B, = U A, = A,
i=l i=l i=l i=l