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,!P…A 1 [ A 2 [ ...†ˆP A j ˆ XP…A j † …additive†: …2:11†XjThese three axioms define a countably additive and nonnegative set functionP(A), A S. As we shall see, they constitute a sufficient set of postulates fromwhich all useful properties of the probability function can be derived. Let usgive below some of these important properties.First, P( ;) ˆ 0. Since S and ; are disjoint, we see from Axiom 3 thatP…S† ˆP…S ‡;†ˆP…S†‡P…;†:It then follows from Axiom 2 that1 ˆ 1 ‡ P…;†orP…;† ˆ 0:Second, if A C, then P(A) P(C). Since A C, one can writeA ‡ B ˆ C;where B is a subset of C and disjoint with A. Axiom 3 then givesP…C† ˆP…A ‡ B† ˆP…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 havejP…A [ B† ˆP…A†‡P…B† P…AB†: …2:12†In order to show this, let us write A [ B in terms of the union of twomutually exclusive events. From the second relation in Equations (2.10),we writeA [ B ˆ A ‡ AB:TLFeBOOK