ma3160RandomVector0809B

# ma3160RandomVector0809B - Random Vector 1 Jointly...

This preview shows pages 1–3. Sign up to view the full content.

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.

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 ∈ ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 30

ma3160RandomVector0809B - Random Vector 1 Jointly...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online