Chapter 7. Joint Distribution & Independence

Chapter 7. Joint Distribution & Independence - 7...

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Unformatted text preview: § 7 Joint distribution and independence § 7.1 Joint distribution 7.1.1 Definition. Let X 1 ,...,X n be random variables defined on the same sample space Ω. The joint distribution function of ( X 1 ,...,X n ) is defined by F ( x 1 ,...,x n ) = P ( X 1 ≤ x 1 ,X 2 ≤ x 2 ,...,X n ≤ x n ) ,-∞ < x 1 ,...,x n < ∞ . 7.1.2 Definition. The distribution function F X i of each X i is called the marginal distribution function of X i (as a member of a sequence of random variables X 1 ,X 2 ,... ). 7.1.3 Definition. For discrete random variables X 1 ,...,X n , the joint mass function of ( X 1 ,...,X n ) is f ( x 1 ,...,x n ) = P ( X 1 = x 1 ,X 2 = x 2 ,...,X n = x n ) ,-∞ < x 1 ,...,x n < ∞ . 7.1.4 Definition. The mass function f X i ( x ) ≡ P ( X i = x ) of each X i is called the marginal mass function of X i (as a member of a sequence of random variables X 1 ,X 2 ,... ). 7.1.5 Definition. Random variables X 1 ,...,X n are (jointly) continuous if their joint distribution function F satisfies F ( x 1 ,...,x n ) = Z x 1-∞ ··· Z x n-∞ f ( u 1 ,...,u n ) du n ··· du 1 , for some nonnegative function f : (-∞ , ∞ ) n → [0 , ∞ ). The function f is called the joint density function (or joint pdf ) of ( X 1 ,...,X n ). 7.1.6 Definition. The pdf of X i is called the marginal pdf of X i (as a member of a sequence of random variables X 1 ,X 2 ,... ). 7.1.7 For discrete X 1 ,...,X n with joint mass function f , • the marginal distribution function of X i can be obtained as F X i ( x ) = X u 1 ··· X u i- 1 X u i ≤ x X u i +1 ··· X u n f ( u 1 ,...,u n ); • the marginal mass function of X i can be obtained as f X i ( x ) = X u 1 ··· X u i- 1 X u i +1 ··· X u n f ( u 1 ,...,u i- 1 ,x,u i ,...,u n ) . 40 7.1.8 For continuous X 1 ,...,X n with joint pdf f , • the marginal distribution function of X i can be obtained as F X i ( x ) = Z ∞ u 1 =-∞ ··· Z ∞ u i- 1 =-∞ Z x u i =-∞ Z ∞ u i +1 =-∞ ··· Z ∞ u n =-∞ f ( u 1 ,...,u n ) du n ··· du 1 ; • the marginal pdf of X i can be obtained as f X i ( x ) = Z ∞-∞ ··· Z ∞-∞ f ( u 1 ,...,u i- 1 ,x,u i +1 ,...,u n ) du n ··· du i +1 du i- 1 ··· du 1 ....
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This note was uploaded on 04/29/2010 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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Chapter 7. Joint Distribution &amp; Independence - 7...

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