Stat 445, spring 2020, Homework assignment 1
Question 1 (problem 4.3 of text)
Let X et N3(μ, Σ) with
and
002 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 and X3 2
04/02/2020
−3 μ=1
4
1 −2 0 Σ=−2 50
d. X2 and X2 − 5 X1 − X3 2
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 using the data
7 H0:μ= 11
2 12 X=89 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.
69 10 6.
83
Verify that T 2 remains unchanged if each observation xj , j = 1, 2, 3 is replaced by C xj and μ0 is replaced by
Cμ0, where
Note that the transformed data matrix is
1 −1 C=11.
(6−9) (6+9) (10−6) (10+6) .
(8−3) (8+3)
2