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Unformatted text preview: Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Lecture 18 Chapter 5: Distribution of Sums and the Central Limit Theorem Michael Akritas Michael Akritas Lecture 18 Chapter 5: Distribution of Sums and the [.2cm] Cen Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums The Normal Case The NonNormal Case The Central Limit Theorem Normal Approximation to the Binomial Michael Akritas Lecture 18 Chapter 5: Distribution of Sums and the [.2cm] Cen Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Michael Akritas Lecture 18 Chapter 5: Distribution of Sums and the [.2cm] Cen Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Definition The function h ( X 1 ,..., X n ) is a linear combination of X 1 ,..., X n , with coefficients a 1 ,..., a n , if h ( X 1 ,..., X n ) = a 1 X 1 + a 2 X 2 + ··· + a n X n . Michael Akritas Lecture 18 Chapter 5: Distribution of Sums and the [.2cm] Cen Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Definition The function h ( X 1 ,..., X n ) is a linear combination of X 1 ,..., X n , with coefficients a 1 ,..., a n , if h ( X 1 ,..., X n ) = a 1 X 1 + a 2 X 2 + ··· + a n X n . Example (The sum and average as linear combinations.) T = X 1 + ··· + X n is a linear combination with all a i = 1. X = 1 n T is a linear combinations with all a i = 1 n . Michael Akritas Lecture 18 Chapter 5: Distribution of Sums and the [.2cm] Cen Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Definition The function h ( X 1 ,..., X n ) is a linear combination of X 1 ,..., X n , with coefficients a 1 ,..., a n , if h ( X 1 ,..., X n ) = a 1 X 1 + a 2 X 2 + ··· + a n X n . Example (The sum and average as linear combinations.) T = X 1 + ··· + X n is a linear combination with all a i = 1. X = 1 n T is a linear combinations with all a i = 1 n . Proposition Let X 1 ,..., X n be any r.v.s (i.e discrete or continuous, independent or dependent). Then E ( a 1 X 1 + ··· + a n X n ) = a 1 μ 1 + ··· + a n μ n , where μ i = E ( X i ) . Michael Akritas Lecture 18 Chapter 5: Distribution of Sums and the [.2cm] Cen Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Corollary 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 . Michael Akritas Lecture 18 Chapter 5: Distribution of Sums and the [.2cm] Cen Outline Expected Value of Sums of R.V.s Variance of Sums Distribution of Sums Corollary 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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This note was uploaded on 03/19/2009 for the course STAT 401 taught by Professor Akritas during the Spring '00 term at Pennsylvania State University, University Park.
 Spring '00
 Akritas
 Central Limit Theorem, Variance

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