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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 6 Review Joint and Marginal Distributions 1. 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 , ) The distribution function F X i of each X i is called the marginal distribution function of X i . 2. For discrete random variables X 1 ,...,X n , the joint probability 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 , ) The mass function f X i = P ( X i = x ) of each X i is called the marginal probability mass function of X i . 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 +1 ,...,u n ) 3. 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 probability density function of ( X 1 ,...,X n ). The pdf of X i is called the marginal pdf of X i . 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 . 4. If a joint distribution function F possesses all partial derivatives at ( X 1 ,...,X n ), then the joint pdf is f ( x 1 ,...,x n ) = ∂ n ∂x 1 ...∂x n F ( x 1 ,...,x n ) . 1 Independence of random variables Random variables X 1 ,...,X n are independent if and only if their joint pmf(pdf) or cdf is equal to the product of their marginal pmfs(pdfs) or cdfs, i.e. f ( x 1 ,...,x n ) = f X 1 ( x 1 ) ...f X n ( x n ) or F ( x 1 ,...,x n ) = F X 1 ( x 1 ) ...F X n ( x n ) Proposition Random variables X and Y are independent if and only if 1. the supports of X and Y do not depend on each other 2. f ( x,y ) can be factorized as g ( x ) h ( y ) This proposition applies to both discrete and continuous random variables and can be generalized to...
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.
- Spring '11