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)