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# hw1 - Stat 771 Fall 2011 Homework 1 Due Wednesday February...

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Stat 771, Fall 2011: Homework 1 Due Wednesday, February 6 1. Let A = 2 1 1 2 , B = 2 - 1 - 1 2 , and c = 1 3 . (a) Find 3 B . (b) Find A - B . (c) Find AB . (d) Find | A | . (e) Find A - 1 . (f) Is A full rank? Why or why not? (g) Let x = x 1 x 2 . Show x 0 Ax = 2( x 2 1 + x 1 x 2 + x 2 2 ) . (h) Use A - 1 to solve the system of two equations in two unknowns 2 x 1 + x 2 = 1 x 1 + 2 x 2 = 3 . That is, solve the system Ax = c . (i) Show tr ( A + B ) = tr ( A ) + tr ( B ). (j) Let Y = Y 1 Y 2 be a random vector with mean μ = μ 1 μ 2 and covariance matrix Σ = σ 2 1 σ 12 σ 12 σ 2 2 . Use results on p. 46 to find E ( AY + c ) and var ( AY + c ). 2. Say Y ij is the j th measurement on subject i , where i = 1 , . . . , n and j = 1 , . . . , 4. All n individuals have the same mean vector (multivariate one-sample problem). Define Y i = Y i 1 Y i 2 Y i 3 Y i 4 and E ( Y i ) = μ = μ 1 μ 2 μ 3 μ 4 . Say we want to show that the mean changes over time. The null hypothesis is that this doesn’t happen H 0 : μ 1 = μ 2 = μ 3 = μ 4 . Find a 3 × 4 matrix C

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hw1 - Stat 771 Fall 2011 Homework 1 Due Wednesday February...

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