GE 331-Lecture 15

GE 331-Lecture 15 - Homework is on compass IE 300/GE 331...

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IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 1 Homework is on compass.

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IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 2 Two discrete r.v.’s • Joint probability mass function of X,Y : f XY (1,2)=P(X=1,Y=2)=0.02, etc. • Requirements: 1 ) , ( 0 1 ) , ( , = y x f y x f XY y x XY ) , ( ) , ( y Y x X P y x f XY = = =
IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 3 Marginal probability distribution • Marginal probability mass function of X • Condition probability mass funciton of Y given X=x = = = y XY X y x f x X P x f ) , ( ) ( ) ( ) ( / ) , ( ) ( / ) , ( ) | ( ) ( | x f y x f x X P x X y Y P x X y Y P y f X XY x Y = = = = = = = =

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IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 4 Independence X and Y are independent if for any x and y f Y|x (y)=f Y (y) or equivalently f XY (x,y)= f X (x)f Y (y) since f Y|x (y)=f XY (x,y)/f X (x)= f Y (y)
IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 5 Two continuous r.v.’s • Joint probability density function f XY (x,y) • Requirements: ∫∫ = A XY dxdy y x f A Y X P ) , ( ) ) , (( ∫∫ + + = 1 ) , ( , 0 ) , ( dxdy y x f y x f XY XY

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IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 6 Example: the joint pdf of X,Y is f XY (x,y)=2(x+y) for 0<x<1 and 0<y<x f=0 anywhere else Joint probability distribution
IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 7 Joint probability distribution (cont) •W h a t i s P(X>1/2, Y<1/2) ∫∫ = + = < > 1 0 2 1 2 1 2 1 2 1 2 / 1 ) ( 2 ) , ( dydx y x Y X P

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IE 300/GE 331 Lecture 15 Negar Kiyavash, UIUC 8 Marginal probability distribution: Mean and Variance • Mean and variance ∫∫ +∞ +∞ +∞ = = = dydx y x xf dx x xf X E XY X X ) , ( ) ( ) ( μ 2 2 2 ) , ( ) ( ) ( ) ( X XY X X dydx y x f x dx x f x X V = = +∞
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This note was uploaded on 09/08/2009 for the course GE 331 taught by Professor Negarkayavash during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE 331-Lecture 15 - Homework is on compass IE 300/GE 331...

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