Chapter 7. Joint Distribution &amp; Independence

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

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Unformatted text preview: Â§ 7 Joint distribution and independence Â§ 7.1 Joint distribution 7.1.1 Definition. Let X 1 ,...,X n be random variables defined on the same sample space Î©. The joint distribution function of ( X 1 ,...,X n ) is defined by F ( x 1 ,...,x n ) = P ( X 1 â‰¤ x 1 ,X 2 â‰¤ x 2 ,...,X n â‰¤ x n ) ,-âˆž < x 1 ,...,x n < âˆž . 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 ) ,-âˆž < x 1 ,...,x n < âˆž . 7.1.4 Definition. The mass function f X i ( x ) â‰¡ 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-âˆž Â·Â·Â· Z x n-âˆž f ( u 1 ,...,u n ) du n Â·Â·Â· du 1 , for some nonnegative function f : (-âˆž , âˆž ) n â†’ [0 , âˆž ). 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 , â€¢ the marginal distribution function of X i can be obtained as F X i ( x ) = X u 1 Â·Â·Â· X u i- 1 X u i â‰¤ x X u i +1 Â·Â·Â· X u n f ( u 1 ,...,u n ); â€¢ the marginal mass function of X i can be obtained as f X i ( x ) = X u 1 Â·Â·Â· X u i- 1 X u i +1 Â·Â·Â· 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 , â€¢ the marginal distribution function of X i can be obtained as F X i ( x ) = Z âˆž u 1 =-âˆž Â·Â·Â· Z âˆž u i- 1 =-âˆž Z x u i =-âˆž Z âˆž u i +1 =-âˆž Â·Â·Â· Z âˆž u n =-âˆž f ( u 1 ,...,u n ) du n Â·Â·Â· du 1 ; â€¢ the marginal pdf of X i can be obtained as f X i ( x ) = Z âˆž-âˆž Â·Â·Â· Z âˆž-âˆž f ( u 1 ,...,u i- 1 ,x,u i +1 ,...,u n ) du n Â·Â·Â· du i +1 du i- 1 Â·Â·Â· 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 - Â 7...

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