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Unformatted text preview: Random Vector 1 Jointly Distributed Random Variables 1.1 Let ( 29 , , P Ω Γ be a probability space. Let : i X R Ω → , 1, , i n = L be random variables. ( 29 1 2 , , , n X X X X = uur L is called a random vector of dimension n . The joint cumulative distribution function (joint cdf) of a random vector ( 29 1 2 , , , n X X X X = uur L of dimension n , is the function 1 , , : n n X X F R R → L given by ( 29 ( 29 { } ( 29 { } ( 29 { } ( 29 ( 29 1 , , 1 2 1 1 2 2 1 1 2 2 , ,..., : : : , ,..., n X X n n n n n F x x x P X x X x X x P X x X x X x ϖ ϖ ϖ ϖ ϖ ϖ ≡ ∈ Ω ≤ ∩ ∈ Ω ≤ ∩ ∩ ∈Ω ≤ ≡ ≤ ≤ ≤ L L 1.2 Joint distributions of two discrete random variables X and Y Let , X Y be discrete random variables. The joint pmf (joint mass) of X and Y is given by the following table: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 , 1 1 , 1 2 , 1 1 2 , 2 1 , 2 2 , 2 , 1 , 2 , , , , , , , , n X Y X Y X Y n X Y X Y X Y n m X Y m X Y m Y y y y X f x y f x y f x y x x f x y f x y f x y x f x y f x y O L L L L L L M M M M M L M M M M ( 29 , , X Y m n f x y M L M M ( 29 ( 29 , , , X Y i j i j f x y P X x Y y = = = satisfies: (a) , ( , ) 0 for 1, , , ; 1, , , X Y i j f x y i m j n ≥ = = L L L L ; (b) ( 29 , , 1 X Y i j i j f x y = ∑∑ ; (c) ( 29 ( 29 { } ( 29 { } { } 1 1 , ( , ) , , , , , , , : ( ), ( ) ( , ) m n X Y x y A x x y y P X Y A P X Y A f x y ϖ ϖ ϖ ∈ ∩ × ∈ ≡ ∈ Ω ∈ = ∑ L L L L , where 2 A R ⊂ . Example * Let X = number of absentees from the morning shift; Y = number of absentees from the evening shift. Based on a long series of past attendance records, the personnel manager provides the assessment of the joint pmf of X (taking 0, 1, 2) and Y (taking 0, 1, 2, 3) as follows: 1 2 3 Row Sum 0.05 0.05 0.1 0.2 0.05 0.1 0.25 0.1 0.5 1 0.15 0.1 0.05 0.3 2 0.1 0.3 0.45 0.15 1 Column Sum Y X O From this we are able to calculate the probability of any event concerning X and Y, for example, ( 29 3 0 0.25 0.15 0.4 P X Y + = = + + = , ( 29 2 0 0.15 0.1 0.05 0.3 P X = = + + + = , 1 ( 29 0.05 0 0.15 0.2 P X Y = + + = . W 1.3 Marginal Probability Mass Functions Let ( 29 , , X Y i j ij f x y p = , ( 29 ( 29 , 1 1 , X i X Y i j ij i j j f x f x y p p ≥ ≥ = = ≡ ∑ ∑ g , 1 i ≥ is called the marginal probability mass function of X and ( 29 ( 29 , 1 1 , Y j X Y i j ij j i i f y f x y p p ≥ ≥ = = ≡ ∑ ∑ g , 1 j ≥ the marginal pmf of Y. Example (Cont'd from *) The marginal pmf of X is given by the row sum column ( 29 1 2 0.2 0.5 0.3 X X f x The marginal pmf of Y is given by the column sum row ( 29 1 2 3 0.1 0.3 0.45 0.15 Y Y f y W Recall that the joint distribution function of X and Y is ( 29 ( 29 , , , X Y F x y P X x Y y = ≤ ≤ for ( 29 2 , x y R ∈ ....
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This note was uploaded on 01/11/2011 for the course MA 3160 taught by Professor Cwwoo during the Spring '07 term at City University of Hong Kong.
 Spring '07
 CWWoo
 Probability

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