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W9-slides2 - Lecture 26 Outline and Examples Expectation...

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Lecture 26- Outline and Examples Expectation, Covariance, Variance and Correlation (Ross § 7.4)
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Three properties of Variance: Property V1: Computational formula for Variance. Var( X ) = E ( X 2 ) - [ E ( X )] 2 . Property V2: Scaling and Shifting. Var( aX + b ) = a 2 Var( X ). Property V3: Var( b ) = 0 for any constant b . Var( X ) = 0 if and only if P ( X = μ ) = 1.
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Example. Ex 4a p. 324-325 Let X 1 , . . . , X n be independent and identically distributed random variables having expected value μ and variance σ 2 . Let ¯ X = P n i =1 X i n be the sample mean. The quantities X i - ¯ X , i = 1 , . . . , n are called deviations , as they equal the differences between the individual data and the sample mean. The random variable S 2 = n X i =1 ( X i - ¯ X ) 2 n - 1 is called the sample variance . Find (a) Var( ¯ X ) and (b) E [ S 2 ]. Solution.
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Example. Ex 4a p. 324-325 Let X 1 , . . . , X n be independent and identically distributed random variables having expected value μ and variance σ 2 . Let ¯ X = P n i =1 X i n be the sample mean. The quantities X i - ¯ X , i = 1 , . . . , n are called deviations , as they equal the differences between the individual data and the sample mean. The random variable
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