{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–12. Sign up to view the full content.

Chapter 5, 6 Multiple Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ω be a given outcome, e.g. Tom H ( ω ) : height of student ω W ( ω ) : weight of student ω A ( ω ) : age of student ω 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ∂x∂y 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 0 a27 A integraldisplay 1 0 integraldisplay x 0 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Pairs of Random Variables Example: f X,Y ( x, y ) = 1 if 0 x 1 , 0 y 1 0 otherwise. Find F X,Y ( x, y ) 1) x 0 or y 0 , F X,Y ( x, y ) = 0 2) 0 x 1 , and 0 y 1 F X,Y ( x, y ) = integraldisplay x 0 integraldisplay y 0 1 d y prime d x prime = xy 3) 0 x 1 , and y > 1 F X,Y ( x, y ) = integraldisplay x 0 integraldisplay 1 0 1 d y prime d x prime = x 4) x > 1 and 0 y < 1 F X,Y ( x, y ) = y 5) x > 1 and y > 1 F X,Y ( x, y ) = 1 ENCS6161 – p.9/47
Independence P X,Y ( x j , y k ) = P X ( x j ) P Y ( y k ) , for all x j and y k (discrete r.v.s) or F X,Y ( x, y ) = F X ( x ) F Y ( y ) for all x and y or f X,Y ( x, y ) = f X ( x ) f Y ( y )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 48

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

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online