b.lect14 - Outline Mean Value of Functions of Random...

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Outline Mean Value of Functions of Random Variables Lesson 14 Chapter 4: Multivariate Variables and Their Distribution M. George Akritas M. George Akritas Lesson 14 Chapter 4: Multivariate Variables and Their Distribu
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Outline Mean Value of Functions of Random Variables Mean Value of Functions of Random Variables The Basic Result Expected Value of Sums Variance of Sums M. George Akritas Lesson 14 Chapter 4: Multivariate Variables and Their Distribu
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Outline Mean Value of Functions of Random Variables The Basic Result Expected Value of Sums Variance of Sums I As in the univariate case the expected value and, consequently, the variance of a function of random variables (statistic) can be obtained without having to first obtain its distribution. Proposition I E ( h ( X , Y )) = X x X y h ( x , y ) p X , Y ( x , y ) I σ 2 h ( X , Y ) = E [ h 2 ( X , Y )] - [ E ( h ( X , Y ))] 2 Formulas for the continuous case are given in Proposition 4.4.1. M. George Akritas Lesson 14 Chapter 4: Multivariate Variables and Their Distribu
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Outline Mean Value of Functions of Random Variables The Basic Result Expected Value of Sums Variance of Sums Example Find E ( X + Y ) and σ 2 X + Y if the joint pmf of ( X , Y ) is y p ( x , y ) 0 1 2 0 0 . 10 0 . 04 0 . 02 x 1 0 . 08 0 . 20 0 . 06 2 0 . 06 0 . 14 0 . 30 (2.1) Solution: By the previous formula, E ( X + Y ) = (0)0 . 1 + (1)0 . 04 + (2)0 . 02 + (1)0 . 08 + (2)0 . 2 + (3)0 . 06 + (2)0 . 06 + (3)0 . 14 + (4)0 . 3 = 2 . 48 . Next M. George Akritas Lesson 14 Chapter 4: Multivariate Variables and Their Distribu
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Outline Mean Value of Functions of Random Variables The Basic Result Expected Value of Sums Variance of Sums Example (Continued) E [( X + Y ) 2 ] = (0)0 . 1 + (1)0 . 04 + (2 2 )0 . 02 + (1)0 . 08 + (2 2 )0 . 2 + (3 2 )0 . 06 + (2 2 )0 . 06 + (3 2 )0 . 14 + (4 2 )0 . 3 = 7 . 66 . Thus σ 2 X + Y = 7 . 66 - 2 . 48 2 = 1 . 51. Example Find E (min { X , Y } ) and Var(min { X , Y } ) if the joint pmf of ( X , Y ) is as in the previous example. M. George Akritas Lesson 14 Chapter 4: Multivariate Variables and Their Distribu
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Outline Mean Value of Functions of Random Variables The Basic Result Expected Value of Sums Variance of Sums Proposition If X and Y are independent, then E ( g ( X ) h ( Y )) = E ( g ( X )) E ( h ( Y )) holds for any functions g ( x ) and h ( y ) . M. George Akritas Lesson 14 Chapter 4: Multivariate Variables and Their Distribu
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Outline Mean Value of Functions of Random Variables The Basic Result Expected Value of Sums Variance of Sums I h ( X 1 , . . . , X n ) is a linear combination of X 1 , . . . , X n if h ( X 1 , . . . , X n ) = a 1 X 1 + · · · + a n X n . I X is a linear combination with all a i = 1 / n . I T = n i =1 X i is a linear combination with all a i = 1. Proposition Let X 1 , . . . , X n be any r.v.s (i.e discrete or continuous, independent or dependent), with E ( X i ) = μ i . Then E ( a 1 X 1 + · · · + a n X n ) = a 1 μ 1 + · · · + a n μ n . M. George Akritas Lesson 14 Chapter 4: Multivariate Variables and Their Distribu
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Outline Mean Value of Functions of Random Variables The Basic Result Expected Value of Sums Variance of Sums Corollary 1. Let X 1 , X 2 be any two r.v.s. Then E ( X 1 - X 2 ) = μ 1 - μ 2 , and E ( X 1 + X 2 ) = μ 1 + μ 2 .
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