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日期:2020-02-12 08:38


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+X2 2

and X3

d. X2 and X2 52X1 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 Xˉ

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

c. The distribution of n(Xˉ μ)T Σ 1(Xˉ μ)

d. The distribution of n(Xˉ μ)T S 1(Xˉ 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. Xˉ

b. n(Xˉ μ)T S 1(Xˉ 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 T2

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.

??

6 9

10 6

8 3

?? .

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

??

(6 9) (6 + 9)

(10 6) (10 + 6)

(8 3) (8 + 3)

?? . 2


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