07_02 - Proof n(n 1)S 2 = i=1(Xi(X n]2 n n 2 2 = i=1(Xi...

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p. 7-11 Proof . Note. The previous three corollaries also hold if X 1 , …, X n are independent. ( n 1) S 2 = n i =1 [( X i μ ) ( X n μ )] 2 = n i ( X i μ ) 2 + n i ( X n μ ) 2 2( X n μ )[ n i ( X i μ )] = n i ( X i μ ) 2 + n ( X n μ ) 2 2 n ( X n μ ) 2 = n i ( X i μ ) 2 n ( X n μ ) 2 . Therefore, Theorem ( of linear transformation). Cor ( a 0 + a 1 X 1 , b 0 + b 1 Y 1 )=sign( a 1 b 1 ) × Cor ( X 1 , Y 1 ), and | Cor ( a 0 + a 1 X , b 0 + b 1 Y )|=| Cor ( X , Y )|, i.e.,| XY | is invariant under location and scale changes. p. 7-12 Theorem (some properties of correlation coefficient).   1 XY 1. (2) XY = 1 if and only if P ( Y = aX + b )=1. (3) Furthermore, XY =1, if a >0 and XY = 1, if a <0. Proof of (1). Proof . Let S = a 0 + a 1 X 1 and T = b 0 + b 1 Y 1 , then Cov ( S , T )= Cov ( a 0 + a 1 X 1 , b 0 + b 1 Y 1 )= a 1 b 1 Cov ( X 1 , Y 1 ), Var ( S )= a 1 2 Var ( X 1 ), and Var ( T )= b 1 2 Var ( Y 1 ). Therefore, ρ ST = Cov ( S, T ) σ S σ T = a 1 b 1 ( X 1 ,Y 1 ) | a 1 || b 1 | σ X σ Y = a 1 b 1 | a 1 b 1 | ρ XY . X Y 0 Var X σ X + Y σ Y = X σ X + Y σ Y +2 X σ X , Y σ Y = ( X ) σ 2 X + ( Y ) σ 2 Y ( X,Y ) σ X σ Y + 1 + 2 ρ ρ ≥− 1 . Similarly, 0 X σ X Y σ Y =1+1 2 ρ ρ 1 .
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p. 7-13 Proof of (2) and (3). We see from the proof of (1), ρ XY =1 Var X σ X Y σ Y =0 . P X σ X Y σ Y = c , where c is a constant. P Y = σ Y σ X X + c σ Y . Similarly, ρ = 1 P Y = σ Y σ X X + c σ Y . Q : How to use expectations to (roughly) characterize random variables X 1 , …, X n ? g ( X 1 , … , X n )= X i E [ g ( X )]= : mean of X i . g ( X 1 , … , X n )=( X i ) 2 E [ g ( X )]= : variance of X i .
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07_02 - Proof n(n 1)S 2 = i=1(Xi(X n]2 n n 2 2 = i=1(Xi...

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