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Example_class_8_handout_solution0

Example_class_8_handout_solution0 - THE UNIVERSITY OF HONG...

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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 function of ( X 1 , . . . , X n ) . The pdf of X i is called the marginal pdf of X i . f X i ( x ) = ˆ -∞ . . . ˆ -∞ 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 ) . 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 multivariate cases. 1

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Expectation of function of random variables Definition For random variables X 1 , . . . , X n with joint pmf or pdf f ( x 1 , . . . , x n ) , if u ( X 1 , . . . , X n ) is a function of these random variables, then the expectation of this function is defined as E ( u ( X 1 , . . . , X n )) = X x 1 . . .
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