EXERCISE 4.1
Assume that the random
vector

has the following normal distribution:

. Transform it according to (
4.49)
to create

with mean

and

.
How would you implement the resulting formula on a computer?
EXERCISE 4.2
Prove Theorem
4.7 using Theorem
4.5.
EXERCISE 4.3
Suppose that

has mean zero and covariance

. Let

. Write

as a linear
transformation, i.e., find the transformation matrix

.
Then compute

via (
4.26). Can you obtain the
result in another fashion?
EXERCISE 4.5
Compute the conditional moments

and

for the pdf of Example
4.5.
EXERCISE 4.6
Prove the relation (
4.28).
EXERCISE 4.7
Prove the relation (
4.29).
Hint: Note that

and that
![$E(\Var(X_2\vert X_1)) = E[E(X_2X_2^{\top}\vert X_1) - E(X_2\vert X_1) \, E(X_2^{\top}\vert X_1)]$](mvahtmlimg1578.gif)
.
EXERCISE 4.8
Compute (
4.46) for the pdf of Example
4.5.
EXERCISE 4.10
Compute (
4.46) for a two-dimensional standard normal
distribution. Show that the transformed random variables

and

are independent. Give a geometrical interpretation of this
result based on iso-distance curves.
EXERCISE 4.11
Consider the Cauchy distribution which has no moment, so that
the CLT cannot be applied. Simulate the distribution of

(for different

's). What can you expect for

?
Hint: The Cauchy distribution can be simulated by the quotient of
two independent standard normally distributed random variables.
EXERCISE 4.13
Consider the pdf

and let

and

. Compute

.
EXERCISE 4.14
Consider the pdf`s
For each of these pdf`s compute

and

EXERCISE 4.15
Consider the pdf

.
Compute

and

EXERCISE 4.16
Consider the pdf
Let

and

.
Compute

.
EXERCISE 4.19
Consider the pdf
Compute

and

.
Also give the best approximation of

by a function
of

. Compute the variance of the error of the approximation.
EXERCISE 4.20
Prove Theorem
4.6.