# homework2 - PSTAT 220C - Advanced Statistical Methods Tony...

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Homework 2 PSTAT 220C - Advanced Statistical Methods Tony Pourmohamad Problem 4.2 Consider the bivariate normal population with μ 1 = 0 2 = 2 11 = 2 22 = 1 , and ρ 12 = . 5 . Part A : Write out the bivariate normal density. f ( x 1 ,x 2 ) = 1 2 π p σ 11 σ 22 (1 - ρ 2 12 ) · exp ( - 1 2(1 - ρ 2 12 ) " ± x 1 - μ 1 σ 11 ² 2 + ± x 2 - μ 2 σ 22 ² 2 - 2 ρ 12 ± x 1 - μ 1 σ 11 ²± x 2 - μ 2 σ 22 ² #) = 1 2 π p 2 · 1(1 - . 5 2 ) · exp ( - 1 2(1 - . 5 2 ) " ± x 1 - 0 2 ² 2 + ± x 2 - 2 1 ² 2 - 2 . 5 ± x 1 - 0 2 ²± x 2 - 2 1 ² #) = 1 2 π q 2 3 · exp ³ - 2 3 ´ x 2 1 2 + ( x 2 - 2) 2 - x 1 ( x 2 - 2) 2 µ¶ Part B : Write out the squared generalized distance expression ( x - μ ) 0 Σ - 1 ( x - μ ) as a function of x 1 and x 2 . ( x - μ ) 0 Σ - 1 ( x - μ ) = 1 1 - ρ 2 12 " ± x 1 - μ 1 σ 11 ² 2 + ± x 2 - μ 2 σ 22 ² 2 - 2 ρ 12 ± x 1 - μ 1 σ 11 ²± x 2 - μ 2 σ 22 ² # = 1 1 - . 5 2 " ± x 1 - 0 2 ² 2 + ± x 2 - 2 1 ² 2 - 2( . 5) ± x 1 - 0 2 ²± x 2 - 2 1 ² # = 4 3 ´ x 2 1 2 + ( x 2 - 2) 2 - x 1 ( x 2 - 2) 2 µ Part C : Determine (and sketch) the constant-density contour that contains 50% of the probability. Problem 4.3 Let X be N 3 ( μ, Σ ) with μ 0 = [ - 3 , 1 , 4] and Σ = 1 - 2 0 - 2 5 0 0 0 2 . Which of the following random variables are independent? Explain. Part A : X 1 and X 2 X 1 and X 2 are not independent since σ 12 = - 2. Part B : X 2 and X 3 X 2 and X 3 are not independent since σ 23 = 0. Part C : ( X 1 ,X 2 ) and X 3 Part D: X 1 + X 2 2 and X 3 Part E : X 2 and X 2 - 5 2 X 1 - X 3 1

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Unformatted text preview: PSTAT 220C - Advanced Statistical Methods Tony Pourmohamad Problem 4.21 Part A : Let X 1 ,..., X 60 be a random sample of size 60 from a four-variate normal distribution having mean μ and covariance matrix Σ . What is the approximate distribution of each of the following? Part A : The distribution of ¯ X ¯ X ∼ N 4 ± μ, 1 60 Σ ² Part B : The distribution of ( X 1-μ ) Σ-1 ( X 1-μ ) ( X 1-μ ) Σ-1 ( X 1-μ ) ∼ Part C : The distribution of n ( ¯ X-μ ) Σ-1 ( ¯ X-μ ) n ( ¯ X-μ ) Σ-1 ( ¯ X-μ ) ∼ χ 2 4 Part D : The approximate distribution of n ( ¯ X-μ ) S-1 ( ¯ X-μ ) n ( ¯ X-μ ) S-1 ( ¯ X-μ ) . ∼ χ 2 4 Problem 5.19 Part A : Part A : Problem 5.20 Part A : Part A : 2...
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## This note was uploaded on 04/16/2010 for the course PSAT 220C taught by Professor Hill during the Spring '06 term at Academy of Design Tampa.

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homework2 - PSTAT 220C - Advanced Statistical Methods Tony...

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