” Stat 445, spring 2020, Homework assignment 1
04/02/2020
Question 1 (problem 4.3 of text)
Let X et N3(, ) with
= ?? 314 ??
and
=
?? 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)
辅导Stat 445作业、 写作MATLAB程序语言作业、MATLAB,python设计作业 辅导Let X1, X2, X3 and X4 be independent Np(, ) 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 and
covariance . Specify each of the following completely.
a. The distribution of Xˉ
b. The distribution of (X1 )T 1(X1 )
c. The distribution of n(Xˉ )T 1(Xˉ )
d. The distribution of n(Xˉ )T S 1(Xˉ )
Question 4 (problem 4.22 from the text)
Let X1, . . . , X75 be a random sample from a population distribution with mean and covariance . What is
the approximate distribution of each of the following?
a. Xˉ
b. n(Xˉ )T S 1(Xˉ ) 1
Question 5 (problem 5.1 from the text)
a. Evaluate T2
for testing
H0 : = 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 = 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 0 is replaced by
C0, 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)
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