Page 24 - Probability, Random Variables and Random Processes

16 PROBABILITY [CHAP 1

THE NOTION AND AXIOMS OF PROBABILITY
1.18. Using the axioms of probability, prove Eq. (1.25).
We have
S=AuA and AnA=@
Thus, by axioms 2 and 3, it follows that
P(S) = 1 = P(A) + P(A)
from which we obtain
P(A) = 1 - P(A)

1.19. Verify Eq. (1.26).
From Eq. (1 Z), we have
P(A) = 1 - P(A)
Let A = @. Then, by Eq. (1.2), A = @ = S, and by axiom 2 we obtain
P(@)=l-P(S)=l-1=0

1.20. Verify Eq. (1.27).
Let A c B. Then from the Venn diagram shown in Fig. 1-7, we see that
B=Au(AnB) and An(AnB)=@
Hence, from axiom 3,
P(B) = P(A) + P(A n B)
However, by axiom 1, P(A n B) 2 0. Thus, we conclude that
P(A)IP(B) ifAcB

Shaded region: A n B
Fig. 1-7
1.21. Verify Eq. (1 .29).
From the Venn diagram of Fig. 1-8, each of the sets A u B and B can be represented, respectively, as a
union of mutually exclusive sets as follows:
AuB=Au(An B) and B=(AnB)u(AnB)
Thus, by axiom 3,
P(A u B) = P(A) + P(A n B)
and P(B) = P(A n B) + P(A n B)
From Eq. (l.61), we have
P(A n B) = P(B) - P(A n B)
Substituting Eq. (1.62) into Eq. (1.60), we obtain
P(A u B) = P(A) + P(B) - P(A n B)