STAT1301 Chap3 Jointly Distributed Random Varibles

STAT1301 Chap3 Jointly Distributed Random Varibles -...

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Stat1301B Probability& Statistics I Fall 2010-2011 P.104 Chapter III Jointly Distributed Random Variables § 3.1 Joint and Marginal Distributions When an experiment or survey is conducted, two or more random variables are often observed simultaneously not only to study their individual probabilistic behaviours but also to determine the degree of relationship among the variables as in most of cases, the variables are related. The probabilistic behaviours of the random variables are described by their joint distribution . In the simplest case, suppose there are only two discrete random variables ( X , Y ) which take distinct values : Values of X : ( ) } ,..., , { 2 1 r x x x X = Ω Values of Y : ( ) } ,..., , { 2 1 c y y y Y = Ω Definition The joint probability mass function ( joint pmf ) of the discrete random variables X and Y and is defined by ()( ) y Y x X P y x p = = = , ,, ( ) Ω X x , () Ω Y y . Sometimes the joint pmf can be conveniently presented in the form of a two-way table as Values of Y Values of X 1 y 2 y c y 1 x 1 1 , y x p ( ) 2 1 , y x p c y x p , 1 2 x 1 2 , y x p ( ) 2 2 , y x p c y x p , 2 r x 1 , y x p r ( ) 2 , y x p r c r y x p ,
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Stat1301B Probability& Statistics I Fall 2010-2011 P.105 Example 3.1 Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white, and 5 blue balls. If we let X and Y denote, respectively, the number of red and white balls in the sample, then both X and Y takes values 0, 1, 2, 3 only. The joint pmf of ( X , Y ) can be calculated as () ( ) ( ) 220 10 3 12 3 5 balls blue 3 0 , 0 0 , 0 = = = = = = P Y X P p ( ) ( ) 220 40 3 12 2 5 1 4 blue 2 white 1 1 , 0 1 , 0 = = = = = = P Y X P p ( ) 220 12 3 12 1 4 2 3 white 1 red 2 1 , 2 = = = P p ( ) 0 white 2 red 2 2 , 2 = = P p Base on similar calculations, we have Values of Y Values of X 0 1 2 3 Total 0 0.0454 0.1818 0.1364 0.0182 0.3818 1 0.1364 0.2727 0.0818 0 0.4909 2 0.0682 0.0545 0 0 0.1227 3 0.0045 0 0 0 0.0045 Total 0.2545 0.5091 0.2182 0.0182 1.0000 The above probabilities can be also represented by the following expression: = 3 12 3 5 4 3 , y x y x y x p , 3 , 2 , 1 , 0 = x , 3 , 2 , 1 , 0 = y , 3 + y x . It is called the bivariate hypergeometric distribution .
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Stat1301B Probability& Statistics I Fall 2010-2011 P.106 Conditions for a joint pmf 1. () 1 , 0 y x p for all ( ) ( ) Ω Ω Y y X x , . 2. 1 , = ∑∑ Ω ∈Ω X xY y y x p 3. ( ) = A y x y x p A Y X P , , , where ( ) ( ) Ω × Ω Y X A . Example 3.2 For the joint pmf in above example, obviously p satisfies properties 1 and 2. For the probability that there are same number of red and white balls, ( ) ( ) ( ) ( ) ( ) {} ( ) 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , Y X = = Y X P P ( )( ) 3181 . 0 2727 . 0 0454 . 0 1 , 1 0 , 0 = + = + = p p . For the probability that there are less red balls than white balls, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 , 2 , 3 , 1 , 2 , 1 , 3 , 0 , 2 , 0 , 1 , 0 , = < Y X P Y X P ( ) ( ) ( ) 2 , 1 3 , 0 2 , 0 1 , 0 p p p p + + + = 4182 . 0 0818 . 0 0182 . 0 1364 . 0 1818 . 0 = + + + = We may also compute the probability concerning about X only. For example, ( ) ( ) ( ) 3818 . 0 0182 . 0 1364 . 0 1818 . 0 0454 . 0
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STAT1301 Chap3 Jointly Distributed Random Varibles -...

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