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日期：2020-02-08 10:50

Stat 445, spring 2020, Homework assignment 1

04/02/2020

Question 1 (problem 4.3 of text)

Let X et N3(µ, Σ) with

µ =−314 and

Σ =1 −2 0

−2 5 0

0 0 2

Which of the following random variables are independent? Explain?

a. X1 and X2.

b. X2 and X3 c.(X1, X2) and X3

c. X1+X22and X3

d. X2 and X2 −2X1 − X3

Question 2 (problem 4.16 from the text)

Let X1, X2, X3 and X4 be independent Np(µ, Σ) random vectors.

a. Find the marginal distributions for each of the random vectors

V1 = X1/4 − X2/4 + X3/4 − X4/4

V2 = X1/4 + X2/4 − X3/4 − X4/4

b. Find the joint density of the random vectors V1 and V2 defined in part (a).

Question 3 (problem 4.21 from the text)

Let X1, . . . , X60 be a random sample of size 60 from a four-variate normal distribution with mean µ and

covariance Σ. Specify each of the following completely.

a. The distribution of X¯

b. The distribution of (X1 − µ)T Σ−1(X1 − µ)

c. The distribution of n(X¯ − µ)T Σ−1(X¯ − µ)

d. The distribution of n(X¯ − µ)T S−1(X¯ − µ)

Question 4 (problem 4.22 from the text)

Let X1, . . . , X75 be a random sample from a population distribution with mean µ and covariance Σ. What is

the approximate distribution of each of the following?

a. X¯

b. n(X¯ − µ)T S−1(X¯ − µ)1

Question 5 (problem 5.1 from the text)

a. Evaluate T2

for testing

H0 : µ =7 11 using the data

X =2 12

8 9

6 9

8 10

b. Specify the distribution of T

2

for the situation in (a).

c. Using (a) and (b), test H0 at the α = 0.05 level. What conclusion do you reach?

Question 6 (problem 5.2 from the text)

The data in Example 5.1 are as follows.

Verify that T2

remains unchanged if each observation xj , j = 1, 2, 3 is replaced by Cxj and µ0 is replaced by

Cµ0, where

C =1 −1 1 1 .

Note that the transformed data matrix is.