Page 31 - Fundamentals of Probability and Statistics for Engineers

14 Fundamentals of Probability and Statistics for Engineers
. Axiom 1: P(A) 0 (nonnegative).
. Axiom 2: P(S) ˆ 1 (normed).
. Axiom 3: for a countable collection of mutually exclusive events A 1 , A 2 , .. . in S,

!
X X
P A 1 [ A 2 [ ...†ˆ P A j ˆ P A j † additive†: 2:11†
j j

These three axioms define a countably additive and nonnegative set function
P(A), A S. As we shall see, they constitute a sufficient set of postulates from
which all useful properties of the probability function can be derived. Let us
give below some of these important properties.
First, P( ) ; ˆ 0. Since S and ; are disjoint, we see from Axiom 3 that
P S†ˆ P S ‡;† ˆ P S†‡ P ;†:

It then follows from Axiom 2 that

1 ˆ 1 ‡ P ;†

or

P ;† ˆ 0:

Second, if A C, then P(A) P(C). Since A C, one can write

A ‡ B ˆ C;

where B is a subset of C and disjoint with A. Axiom 3 then gives

P C†ˆ P A ‡ B†ˆ P A†‡ P B†:

Since P(B) 0 as required by Axiom 1, we have the desired result.
Third, given two arbitrary events A and B, we have

P A [ B†ˆ P A†‡ P B† P AB†: 2:12†

In order to show this, let us write A [ B in terms of the union of two
mutually exclusive events. From the second relation in Equations (2.10),
we write

A [ B ˆ A ‡ AB:

TLFeBOOK