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Unformatted text preview: Lecture 3 : Random variables and their distributions 3.1 Random variables Let ( , F ) and ( S, S ) be two measurable spaces. A map X : S is measurable or a random variable (denoted r.v.) if X 1 ( A ) { : X ( ) A } F for all A S One can write { X A } or ( X A ) as a shorthand for { : X ( ) A } = X 1 ( A ). If ( S, S ) = ( R d , R d ), then X is called a ddimensional random vector. Here, R is the Borel  field or the field generated by the open subsets of R n , and R d is the dfold product algebra of R with itself, which will be defined shortly. Perhaps the simplest example of a measurable function is an indicator function of a measurable set. The indicator function of a set F F is defined as 1 F ( ) = ( 1 if F 0 if / F 3.2 Generation of field Let A be a collection of subsets of . The field generated by A , denoted by ( A ), is the smallest field on which contains A , which is the intersection of all fields containing A . Let { X i } i I be a family of mappings of into measurable spaces ( S i , S i ), i I . Here, I 6 = is an arbitrary (possibly uncountable) index set . The field generated by { X i } i I , denoted by ( { X i } i I ), is the smallest  field on with respect to which each X i is measurable. Taking A = S i I { X 1 i ( S ) : S S i } , we can see that this case...
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.
 Spring '09
 Reed

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