Course in Probability Theory

3 .1 GENERAL DEFINITIONS 1 37Specifically, F is given byF(x) = A(( - oo, x]) = 91{X < x) .While the r .v. X determines lc and therefore F, the converse is obviouslyfalse . A family of r .v .'s having the same distribution is said to be "identicallydistributed" .Example 1 . Let (S2, J) be a discrete sample space (see Example 1 of Sec . 2 .2) .Every numerically valued function is an r .v .Example 2 . (91, A, m) .In this case an r .v . is by definition just a Borel measurable function . According tothe usual definition, f on 91 is Borel measurable iff f -' (A3') C ?Z . In particular, thefunction f given by f (co) - w is an r .v . The two r .v .'s co and 1 - co are not identicalbut are identically distributed ; in fact their common distribution is the underlyingmeasure m .Example 3 . (9' A' µ)The definition of a Borel measurable function is not affected, since no measureis involved ; so any such function is an r .v ., whatever the given p .m . µ may be . Asin Example 2, there exists an r .v . with the underlying µ as its p.m. ; see Exercise 3below .We proceed to produce new r .v .'s from given ones .Theorem 3 .1 .4 . If X is an r .v ., f a Borel measurable function [on (R1 A 1 )],then f (X) is an r .v .PROOF . The quickest proof is as follows . Regarding the function f (X) ofco as the "composite mapping" :f ° X : co -* f (X (w)),we have (f ° X) -1 = X -1 ° f -1 and consequently(f ° X)-1 0) = X-1(f-1 W)) C X -1 (A1) CThe reader who is not familiar with operations of this kind is advised to spellout the proof above in the old-fashioned manner, which takes only a littlelonger .We must now discuss the notion of a random vector . This is just a vectoreach of whose components is an r .v . It is sufficient to consider the case of twodimensions, since there is no essential difference in higher dimensions apartfrom complication in notation .