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 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 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.
版权所有:编程辅导网 2021 All Rights Reserved 联系方式:QQ:99515681 微信:codinghelp 电子信箱:99515681@qq.com
免责声明:本站部分内容从网络整理而来,只供参考!如有版权问题可联系本站删除。