Page 14 - Probability, Random Variables and Random Processes

PROBABILITY [CHAP 1

and

B. Axiomatic Definition:
Let S be a finite sample space and A be an event in S. Then in the axiomatic definition, the
, probability P(A) of the event A is a real number assigned to A which satisfies the following three
axioms :
Axiom 1 : P(A) 2 0 (1.21)
Axiom 2: P(S) = 1 (1.22)
Axiom 3: P(A u B) = P(A) + P(B) if A n B = 0 (1.23)
If the sample space S is not finite, then axiom 3 must be modified as follows:
Axiom 3': If A,, A,, . . . is an infinite sequence of mutually exclusive events in S (Ai n Aj = 0
for i # j), then

These axioms satisfy our intuitive notion of probability measure obtained from the notion of relative
frequency.

C. Elementary Properties of Probability:
By using the above axioms, the following useful properties of probability can be obtained:

6. If A,, A,, . . . , A, are n arbitrary events in S, then

- ... (-1)"-'P(A1 n A, n --.n A,) (1.30)
where the sum of the second term is over all distinct pairs of events, that of the third term is over
all distinct triples of events, and so forth.
7. If A,, A,, . . . , A, is a finite sequence of mutually exclusive events in S (Ai n Aj = 0 for i # j),
then

and a similar equality holds for any subcollection of the events.
Note that property 4 can be easily derived from axiom 2 and property 3. Since A c S, we have