Chap5_427 - Chap. 5: Joint Probability Distributions...

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1 Chap. 5: Joint Probability Distributions Probability modeling of several RV‟s We often study relationships among variables. Demand on a system = sum of demands from subscribers (D = S 1 + S 2 + …. + S n ) 2 Stress & strain are related to material properties; random loads; etc. Notation: Sometimes we use X 1 , X 2 ,…., X n Sometimes we use X, Y, Z, etc.
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2 Sec 5.1: Basics First, develop for 2 RV (X and Y) Two Main Cases I. Both RV are discrete II. Both RV are continuous I. (p. 185). Joint Probability Mass Function (pmf) of X and Y is defined for all pairs (x,y) by ) , ( ) and ( ) , ( y Y x X P y Y x X P y x p
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3 pmf must satisfy: for any event A, ) , ( all for 0 ) , ( y x y x p 1 ) , (   x y y x p   A y x y x p A Y X P ) , ( ) , ( ) , (
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4 Joint Probability Table: Table presenting joint probability distribution: Entries: P(X = 2, Y = 3) = .13 P(Y = 3) = .22 + .13 = .35 P(Y = 2 or 3) = .15 + .10 + .35 =.60 y 1 2 3 x 1 .10 .15 .22 2 .30 .10 .13 ) , ( y x p
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5 The marginal pmf X and Y are x Y y X y x p y p y x p x p ) , ( ) ( and ) , ( ) ( y 1 2 3 x 1 .10 .15 .22 .47 2 .30 .10 .13 .53 .40 .25 .35 y 1 2 3 p Y (y) .40 .25 .35 x 1 2 p X (x) .47 .53
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6 II. Both continuous (p. 186) A joint probability density function (pdf) of X and Y is a function f(x,y) such that f(x,y) > 0 everywhere . and  A dxdy y x f A Y X P ) , ( ] ) , [(   1 ) , ( dxdy y x f
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7 pdf f is a surface above the (x,y)-plane A is a set in the (x,y)-plane. is the volume of the region over A under f. (Note: It is not the area of A.) x y f A ] ) , [( A Y X P
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Ex) X and Y have joint PDF f(x,y) = c x y 2 if 0 < x < y < 1 = 0 elsewhere. Find c. First, draw the region where f > 0. (not . 0 y x 1 1 dxdy cxy y 0 2 1 0 1 dxdy cxy y x 0 2 1 dydx cxy x 1 2 1 0
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so, c = 10 Find P(X+Y<1) First, add graph of x + y =1 . 0 y x 1 1 10 / 5 . ] | 5 [. 1 0 4 0 2 1 0 2 0 2 1 0 c dy y c dy x y c dxdy cxy y y   1 5 . 1 0 2 0 2 5 . 0 10 10 ) 1 ( y y dxdy xy dxdy xy Y X P 135 . ) ) 1 (( ) 3 / 10 ( 3 10 10 3 3 5 . 0 1 3 5 . 0 5 . 0 1 2   dx x x x dx y x dydx xy x x x x
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10 Marginal pdf (p. 188) Marginal pdf of X : Marginal pdf of Y : Ex) X and Y have joint PDF f(x,y) = 10 x y 2 if 0 < x < y < 1 , and 0 else. For 0 < y < 1: dy y x f x f X ) , ( ) ( dx y x f y f Y ) , ( ) ( 4 0 2 0 2 5 10 10 ) , ( ) ( y xdx y dx xy dx y x f y f y y Y otherwise. 0 ) ( and y f Y
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11 marginal pdf of Y: marginal pdf of Y: you check Notes: 1. x cannot appear in (y can‟t be in ) 2. You must give the ranges; writing is not enough. Math convention: writing
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Chap5_427 - Chap. 5: Joint Probability Distributions...

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