zz-2 - ECE 6010 Lecture 3 Random Vectors Grimmet...

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Unformatted text preview: ECE 6010 Lecture 3 Random Vectors Grimmet & Stirzaker: Section 4.9 Random Vectors Random vectors are an extension of the bivariate random variables. n r.v.s X 1 , X 2 , . . . , X n define a measurable mapping from an underlying sample space ( , F ) to ( R n , B n ) , where B n is the smallest -field containing all sets of the form { ( x 1 , x 2 , . . . , x n ) : a 1 < x 1 b 1 , a 2 < x 2 b 2 , , a n < x n b n } . Definition 1 The joint distribution of X 1 , . . . , X n is P X 1 X 2 ...X n ( B ) = P ( { : ( X 1 ( ) , X 2 ( ) , . . . , X n ( ) B } for all B B This probability is denoted as P X ( B ) . 2 Definition 2 The joint cumulative distribution function c.d.f. if P X 1 X 2 ...X n ( a 1 , a 2 , . . . , a n ) = P ( X 1 a 1 , . . . , X n a n ) = F X ( a ) , a R n . 2 Definition 3 The joint probability mass function (p.m.f.) is p X ( a ) = P ( X 1 = a 1 , . . . , X n = a n ) 2 Definition 4 The joint probability density function (p.d.f.) f X ( a ) is the function that satisfies F X ( a ) = Z a n Z a n- 1 Z a 1 f X ( x ) dx 1 dx 2 dx n for a continuous random vector. 2 Fact: X 1 , X 2 , . . . , X n are independent if F X or p X or f X factor into products of marginals. Suppose g : ( R n , B n ) ( R n , B n ) is measurable. Then g ( X 1 , . . . , X n ) is a random variable. Law of unconscious statistician: E [ g ( X 1 , . . . , X n )] = ( R R g ( x 1 , . . . , x n ) f X ( x ) d x continuous g ( x i 1 , x i 2 , . . . , x i n ) p X ( x i 1 , x i 2 , . . . , x i n ) discrete ....
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This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.

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zz-2 - ECE 6010 Lecture 3 Random Vectors Grimmet...

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