mid1 2007sol

mid1 2007sol - Quiz 1 Solutions 1. (a) Normal with mean 0...

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Quiz 1 Solutions 1. (a) Normal with mean 0 and variance 3. (b) Cov( Y 1 , Y 2 ) = E [[1 1 1] εε T [1 - . 5 - . 5] T ] - 0 = [1 1 1][1 - . 5 - . 5] T = 0 Therefore, Y 1 and Y 2 are uncorrelated. (c) Since Y 1 and Y 2 are jointly normally distributed and uncorrelated, they must be independent. (d) Note that the variance of Y 2 is 3/2, so Y 2 2 (3 / 2) must be χ 2 (1) . (e) Since Y 1 / 3 is standard normal, and independent of Y 2 , the given quantity must be t -distributed on 1 degree of freedom (using the information from the previous part). (f) P T = P and P 2 = P , by straightforward calculations. tr ( P ) = 2. (g) χ 2 (2) . (h) F (2 , 1) . 2. (a) grain ~ rate summary -.0324 .00316 105.3 1 and 3 (b) ˆ y = 22 . 56 - . 0324 x where y = grain, and x = rate. (c) TRUE (d) 22 . 56 - . 0324(140) ± 3 . 18( . 2497) ± 1 5 + (140 - 100) 2 6250 = 18 . 02 ± . 54 3. (a) tr ( H ) = tr ( X ( X T X ) - 1 X T ) = tr ( X T X ( X T X ) - 1 ) = tr ( I k × k ) = k since X is n × k . tr ( I - H ) = tr ( I ) - tr ( H ) = n - k ( I - H ) 2 = I - H - H + H 2 = I - 2 H + H = I - H since H 2 = X ( X T X ) - 1 X T X ( X T X ) - 1 X T = X ( X T X ) - 1 X T = H . ( I - H ) X = X - X ( X T X ) - 1 X T X = X - X = 0 (b) y - ˆ y = + ε - X ˆ β = + ε - X ( X T X ) - 1 X T -
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