MATH4630 Assignment 1 Solved

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1. For the following questions, you have to clearly show your work.

   1. Find the eigenvalues and eigenvectors for A.
   2. Find the square root of A using Cholesky decomposition. 3. Find the square root of A using spectral decomposition.

2. Is A a positive definite matrix? Why or why not?
3. Use any software to verify your answers in part (a).

Question 1: Let

Question 2: Let A be a (p ⇥ p) matrix and is parti!tioned into

B11 B12 B21 B22

where B11 is a (p1 ⇥p1) matrix, B22 is a (p2 ⇥p2) matrix, and p1 +p2 = p. Assume A11, A22, B11, and B22 are non singular matrices.

1. Denote Ip be the (p ⇥ p) identity matrix. Let AB = Ip. Express Bij in terms of Aij for all i,j = 1,2.
2. Let BA = Ip. Express Bij in terms of Aij for all i,j = 1,2.
3. Show that

be a (p ⇥ p) matrix and is partitioned into B=

!

d. Show that and

|A|=|A ||A A A1A |=|A ||A A A1A | 22 11 12 22 21 11 22 21 11 12

B =A1 +A1A (A A A1A )1A A1 11 11 111222 211112 2111

B =A1 +A1A (A A A1A )1A A1 22 22 222111 122221 1222

A11 A12 A21 A22

A=
where A11 is a (p1 ⇥p1) matrix, A22 is a (p2 ⇥p2) matrix, and p1 +p2 = p. Similarly, let B

1

Question 3: Let
fg2 f22122

X= X1 !⇠N μ1 !, ⌃11 ⌃12 !! Xg p μf ⌃ ⌃

where X1 is a p1-dimensional vector, X2 is a p2-dimensional vector, ⌃11 is a (p1 ⇥p1) matrix, gg

⌃22 is a (p2 ⇥ p2) matrix, and p1 + p2 = p. Show that the conditional mean and variance of X1 given X2 = x2 are

ggf

respectively.
Question 4: Consider the following data set:

x1: 3 3 4 5 6 8 x2: 17.95 15.54 14.00 12.95 8.94 7.49

For the following questions, you have to clearly show your steps. Computer commanda and

μ+⌃⌃1(xμ) and ⌃⌃⌃1⌃ f1 12 22 f2 f2 11 12 22 21

print

out is not accepted.

1. Find the sample mean vector.
2. Find the sample unbiased variance matrix.
3. Report the squared statistical distances (xj x ̄)0S1(xj x ̄) for j = 1, . . . , 6.

   eeee

4. Assume the data set is from a bivariate normal distribution.

   1. Describe how you would estimate the 50% probability contour of the population mean vector.

   2. At 5% level of significance, is there significant evidence that the population mean vector is di↵erent from (3, 10)0.

2

Question 5: Data are given in the excel file.

1. Using a graphical method to check if the data of East is a sample from the normal

   distribution. How about data of South, West, and North?

2. Regardless of your result in part (a), obtain the 95% confidence interval for the mean of (1) North (2) South (3) East (4) West .

   Clearly state the necessary assumptions needed for your answer.

3. Considering the data set as a multivariate data set. Use a software and report the sample mean vector, sample covariance matrix and sample correlation matrix.
4. Use a graphical method to check if the data set is a sample from a multivariate normal distribution.
5. Obtain the equation for obtaining the 95% confidence region for the population mean vector, μ = (μN,μS,μE,μW)0. (No calculations needed. Just the equations.) Clearly state the necessary assumptions needed for your answer.
6. At 5% level of significance, test
   H0 : μ = (1450, 1900, 1700, 1700)0 vs Ha : μ 6= (1450, 1900, 1700, 1700)0.

   eee0

7. Based on the your answer in part (f), is μ = (1450, 1900, 1700, 1700) falls within the

   95% confidence region of μe obtained in part (e)? Why or why not?