Stat 771, Fall 2011: Homework 1
Due Wednesday, February 6
1. Let
A
=
•
2
1
1
2
‚
,
B
=
•
2

1

1
2
‚
,
and
c
=
•
1
3
‚
.
(a) Find 3
B
.
(b) Find
A

B
.
(c) Find
AB
.
(d) Find

A

.
(e) Find
A

1
.
(f) Is
A
full rank? Why or why not?
(g) Let
x
=
•
x
1
x
2
‚
. Show
x
0
Ax
= 2(
x
2
1
+
x
1
x
2
+
x
2
2
)
.
(h) Use
A

1
to solve the system of two equations in two unknowns
‰
2
x
1
+
x
2
=
1
x
1
+
2
x
2
=
3
.
That is, solve the system
Ax
=
c
.
(i) Show
tr
(
A
+
B
) =
tr
(
A
) +
tr
(
B
).
(j) Let
Y
=
•
Y
1
Y
2
‚
be a
random vector
with mean
μ
=
•
μ
1
μ
2
‚
and covariance matrix
Σ
=
•
σ
2
1
σ
12
σ
12
σ
2
2
‚
. Use results on p. 46 to find
E
(
AY
+
c
) and
var
(
AY
+
c
).
2. Say
Y
ij
is the
j
th
measurement on subject
i
, where
i
= 1
, . . . , n
and
j
= 1
, . . . ,
4. All
n
individuals
have the same mean vector (multivariate onesample problem). Define
Y
i
=
Y
i
1
Y
i
2
Y
i
3
Y
i
4
and
E
(
Y
i
) =
μ
=
μ
1
μ
2
μ
3
μ
4
.
Say we want to show that the mean changes over time. The null hypothesis is that
this doesn’t happen
H
0
:
μ
1
=
μ
2
=
μ
3
=
μ
4
. Find a 3
×
4 matrix
C
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 Spring '10
 Hanson
 Mean, Means, Yij, Formally test, th measurement

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