ch56 - Chapter 5, 6 Multiple Random Variables ENCS6161 -...

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Unformatted text preview: Chapter 5, 6 Multiple Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University Vector Random Variables A vector r.v. X is a function X : S R n , where S is the sample space of a random experiment. Example: randomly pick up a student name from a list. S = { all student names on the list } . Let be a given outcome, e.g. Tom H ( ) : height of student W ( ) : weight of student A ( ) : age of student H,W,A are r.v.s. Let X = ( H,W,A ) , then X is a vector r.v. ENCS6161 p.1/47 Events Each event involving X = ( X 1 ,X 2 , ,X n ) has a corresponding region in R n . Example: X = ( X 1 ,X 2 ) is a two-dimensional r.v. A = { X 1 + X 2 10 } B = { min( X 1 ,X 2 ) 5 } C = { X 2 1 + X 2 2 100 } ENCS6161 p.2/47 Pairs of Random Variables Pairs of discrete random variables Joint probability mass function P X,Y ( x j ,y k ) = P { X = x j intersectiondisplay Y = y k } = P { X = x j ,Y = y k } Obviously j k P X,Y ( x j ,y k ) = 1 . Marginal Probability Mass Function P X ( x j ) = P { X = x j } = P { X = x j ,Y = anything } = summationdisplay k =1 P X,Y ( x j ,y k ) Similarly P Y ( y k ) = j =1 P X,Y ( x j ,y k ) . ENCS6161 p.3/47 Pairs of Random Variables The joint CDF of X and Y (for both discrete and continuous r.v.s) F X,Y ( x,y ) = P { X x,Y y } a45 a54 x (x,y) y ENCS6161 p.4/47 Pairs of Random Variables Properties of the joint CDF: 1. F X,Y ( x 1 ,y 1 ) F X,Y ( x 2 ,y 2 ) , if x 1 x 2 ,y 1 y 2 . 2. F X,Y (- ,y ) = F X,Y ( x,- ) = 0 3. F X,Y ( , ) = 1 4. F X ( x ) = P { X x } = P { X x,Y = anything } = P { X x,Y } = F X,Y ( X, ) F Y ( y ) = F X,Y ( ,y ) F X ( x ) ,F Y ( y ) : Marginal cdf ENCS6161 p.5/47 Pairs of Random Variables The joint pdf of two jointly continuous r.v.s. f X,Y ( x,y ) = 2 xy F X,Y ( x,y ) Obviously, integraldisplay - integraldisplay - f X,Y ( x,y )d x d y = 1 and F X,Y ( x,y ) = integraldisplay x- integraldisplay y- f X,Y ( x prime ,y prime )d y prime d x prime ENCS6161 p.6/47 Pairs of Random Variables The probability P { a X b,c Y d } = integraldisplay b a integraldisplay d c f X,Y ( x,y )d y d x In general, P { ( X,Y ) A } = integraldisplay integraldisplay A f X,Y ( x,y )d x d y Example: a45 a54 x y 1 1 a27 A integraldisplay 1 integraldisplay x f X,Y ( x prime ,y prime )d y prime d x prime ENCS6161 p.7/47 Pairs of Random Variables Marginal pdf: f X ( x ) = d d x F X ( x ) = d d x F X,Y ( x, ) = d d x parenleftbigg integraldisplay x- integraldisplay - f X,Y ( x prime ,y prime )d y prime d x prime parenrightbigg = integraldisplay - f X,Y ( x,y prime )d y prime f Y ( y ) = integraldisplay - f X,Y ( x prime ,y )d x prime ENCS6161 p.8/47 Pairs of Random Variables Example: f X,Y ( x,y ) = 1 if x 1 , y 1 otherwise....
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This note was uploaded on 01/15/2011 for the course ECE 616 taught by Professor Khkjk during the Winter '10 term at Concordia Canada.

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ch56 - Chapter 5, 6 Multiple Random Variables ENCS6161 -...

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