EXERCISE 5.3
Prove proposition (c) of Theorem 5.7.
EXERCISE 5.4
Let
and
a)
Determine the distribution of .
b)
Determine the distribution of .
EXERCISE 5.5
Consider
Compute and knowing that
Determine the distributions of and of .
EXERCISE 5.6
Knowing that
a)
find the distribution of
and of .
b)
find the distribution of
c)
compute
.
EXERCISE 5.7
Suppose
with positive definite. Is it possible that
a)
,
b)
,
c)
, and
d)
?
EXERCISE 5.8
Let
.
a)
Find the best linear approximation of by a linear function of and and compute the multiple correlation between and .
b)
Let
and
. Compute the distribution of
.
EXERCISE 5.9
Let
be a trivariate normal r.v. with
Find the distribution of
and compute the partial correlation
between and for fixed .
Do you think it is reasonable to approximate by a
linear function of and ?
EXERCISE 5.10
Let
a)
Give the best linear approximation of as a function of and evaluate the quality of the approximation.
b)
Give the best linear approximation of as a function of and
compare your answer with part a).
EXERCISE 5.11
Prove Theorem 5.2.
(Hint: complete the linear transformation
and then use Theorem 5.1 to get the marginal of the first components of .)