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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 8 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 ) = ˆ x 1-∞ ... ˆ 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...
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