Page 88 - Fundamentals of Communications Systems

3.2 Chapter Three

Definition 3.1 Events A and B are mutually exclusive if A∩ B =∅
Axioms of Probability. A probability measure, P , for a probability space
( , F, P ) with events A, B ∈ F must satisfy the following axioms

■ For any event A, P (A) ≥ 0
■ P ( ) = 1
■ If A and B are mutually exclusive events then P (A∪ B) = P (A) + P (B).

These three axioms are the building blocks for probability theory and an
understanding of random events that characterize communication system
performance.

EXAMPLE 3.2
Example 3.1(cont.). The rolling of a fair die

A 1 ={1} A 2 ={2, 4, 6} A 3 ={1, 2, 3} A 4 ={1, 3, 5}
P [{1}] = P [{2}] = ··· = P [{6}] (3.3)
1 1 1 1
P [A 1 ] = P [A 2 ] = P [A 3 ] = P [A 4 ] =
6 2 2 2

C
Definition 3.2 (Complement) The complement of a set A, denoted A , is the set of all
elements of that are not elements of A.

Theorem 3.1 Poincare For N events A 1 , A 2 , .....A N
P [A 1 ∪ A 2 ∪ .... ∪ A N ] = S 1 − S 2 +· · ··+(−1) N −1 S N
where

S k = P [A i 1 ∩ A i 2 ∩ .... ∩ A i k ] i 1 ≥ 1, i k ≤ N
i 1 <i 2 <....<i k
Proof: The results are given for N = 2 and the proof for other cases is similar
C
(if not more tedious). Note that events A and B ∩ A are mutually exclusive
events. Consequently we have
C
C
P [A∪ B] = P [(A∩ B ) ∪ B] = P [A∩ B ] + P [B] (3.4)
C
C
P [A∪ B] = P [(B ∩ A ) ∪ A] = P [B ∩ A ] + P [A] (3.5)
C
C
P [A∪ B] = P [(A∩ B ) ∪ (A∩ B) ∪ (B ∩ A )]
C
C
= P [A∩ B ] + P [A∩ B] + P [B ∩ A ] (3.6)
Using Eq. (3.4) + Eq. (3.5) − Eq. (3.6) gives the desired result.