INTRODUCTION TO STATISTICS AND ECONOMETRICS
Bayes’ Theorem
Bayes’ theorem follows easily from the rules of probability but is listed separately here because of its special usefulness.
THEOREM 2.4.2 (Bayes) Let events A, A2, . . . , An be mutually exclusive such that P{A U A2 U. . . U An) = 1 and Р(Д) > 0 for each i. Let E be an arbitrary event such that P(E) > 0. Then
Proof. From Theorem 2.4.1, we have
Since E П A], E П А4, . . . , E fl An are mutually exclusive and their union is equal to E, we have, from axiom (3) of probability,
(2.4.4) P(E) = X P(E n Aj)-
j= і
Thus the theorem follows from (2.4.3) and (2.4.4) and by noting that P(E П Aj) = P(£ I Aj)P(Aj) by Theorem 2.4.1. □