Math 439 Homework 2

Math 439 Homework 2 - Fall 2008

HOMEWORK #2 due 9-30

In the following, _ means subscript and ^ means superscript.

NOTE: In all problem sets that use SAS, arrange your answers into three parts, in the following order:

(I) Your answers to all questions, either written by hand or using a word processor,

(II) The SAS program source files (*.sas files) that you used in the problem set,

(III) The output from the SAS programs in Part II.

In Part I, you can refer to plots or tables or large matrices that problems ask for by saying (for example), ``The scatterplot or matrix for Problem 3 is on page 17 of the SAS output.'' If necessary, add page numbers to the SAS output, so that (for example) you don't have several different page 1s in Part III.

1. Consider the matrix

    A = (  3   2  )
        (  2   6  )

(i) Find two linearly-independent eigenvectors v_1,v_2 in R^2 and their eigenvalues d_1,d_2. (Hint: One way to start is to solve det(A-zI)=0 and then find vectors v such that (A-zI)v=0.)

(ii) Find an orthogonal matrix O and diagonal matrix D such that A = ODO'. Verify that O an is orthogonal matrix, and verify that ODO'=A. (Hint: This is the spectral theorem for A. Try O=(v_1 v_2). )

(iii) Use part (ii) to find a symmetric commuting square root B of A. That is, find a 2 x 2 matrix B such that B=B', AB=BA, and B^2=A.

2. Consider the matrix

    A = (  1   2   3  )
        (  2   5   7  )
        (  3   7  12  )

Find an upper-triangular matrix B such that A=B'B, and verify that A=B'B. (Hint: See pages 25-26 in the text.)

3. (i) Let X_1 and X_2 be real-valued random variables. Show that

      Cov(X_1-X_2,X_1+X_2) = Var(X_1) - Var(X_2)

(ii) Let A=A' be a 2 x 2 symmetric matrix with tr(A)>0 and det(A)>0. Prove that A is positive definite.

(Hint: Use the spectral decomposition of A.)

4. Let X = (X₁ X₂ X₃)' be a random vector in R³ with mean vector E(X)=0 and covariance matrix

              (  3   -4    1 )
    Cov(X) =  ( -4   10   -2 )
              (  1   -2    3 )

Let Y = X₁ + 2X₂ + 3X₃ and Z = X₂ +4X₃ . Recall that Var(Z) means the variance of Z.

(i) Find Var(X₁) and Var(X₂).

(ii) Find Var(Y) and Var(Z) .

(iii) Find the covariance Cov(Y,Z) = E(YZ) .

(iv) Let W be the random vector (X_1 X_3)'. Find Cov(W).

5. Consider the blood glucose measurements for three visits by each of 50 female subjects in Table 3.8 on page 80 in the text. (See BloodGlucose.dat on the Math 439 Web site.)

(i) Write a SAS program that uses SAS's matrix language (Proc IML) to compute the sample mean vector and sample covariance matrix of the six variables in the table. Partition the mean vector and sample covariance matrix into two groups with three variables each, as in Table 3.8 and Section 3.8 in the text.

Did you get the same answer as in the back of the book? (This problem is similar to problem 3.22 in the text, whose answer is in the back of the book.)

(ii) Have your SAS program also calculate the sample correlation matrix of the six variables.

(iii) In the same program, also use SAS's Proc Corr to compute the sample mean vector, sample covariance matrix, and sample correlation matrix. Do you get the same answers?

(Hint: See Calcium.sas or Calcium2.sas on the Math439 Web site.)

(iv) Which set of three variables (either Before=y1,y2,y3 or After=x1,x2,x3) appear to be more highly correlated among themselves for the 50 subjects? Does this suggest that fasting blood-glucose levels are more uniform among these subjects, or that blood-glucose levels after sugar intake are more uniform?

6. Do problem 4.16 (page 108) with X, mu, and Sigma given by

        ( 1 )          ( -2 )	            (  5   3  |  -1  -1 )
  xx =  ( 2 )    mu =  (  3 )     Sigma =   (  3   7  |   0  -1 )
                       (----)               ( ----------------- )
                       ( -1 )               ( -1   0  |   2   3 ) 
                       (  5 )               ( -1  -1  |   3   5 )

Either do this either by hand, or else include the calculations in your SAS program for Problem 5. (Warning: In the book's notation, the upper-left corner of Sigma is S_yy and the upper-right corner is S_yx, which is the transpose of S_xy. Note that the answer to problem 4.16 is also given in the back of the book.)

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