Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 4 Solutions

Suppose
.

Show that
and
.
Using characteristic functions:

Show that
.
We note that . Then

Suppose
and write
. Show that
.
If we have, by applying the previous result

Show that
and
.

Suppose you have a random number generator which is capable of
generating random numbers distributed as
.
Describe how to generate random vectors
.
Generate dimensional random vectors by generating realizations of the scalar random variable . Then .
Write (the Cholesky factorization). Let . Then

Suppose
and
are r.v.s. Show that
is
minimized over all functions
be the function

First approach: We will assume (only for convenience) that the
r.v.s are continuous. We can write

Approach 2: This one is more appealing, because it is not
expressed in terms of integrals (making it immediately applicable to
all types of distributions), it does not require interchanging
limiting operations (i.e., taking derivatives inside integrals), and
does not even require derivatives with respect to functions (which
was only partially justified in the first approach). This
presentation is due to Ross.
The proof is accomplished by establishing the following inequality:

First approach: We will assume (only for convenience) that the
r.v.s are continuous. We can write

Let
.
Let
, where
First observe that . Write down two expressions for the density, first in terms of the separate parameters,

Suppose
where

The value
is measured. Determine the best estimate
for
.
We can write for the mean of the unmeasured variables. The estimation formula is

In a separate problem, the values
and
are
measured. Determine the best estimate of
.

Determine a random vector
which is a whitened version of
.
Using M ATLAB ,
S = [4 2 1; 2 6 3; 1 3 8]; R = chol(S); C = R'; C*C' % check the result
we determine that

The value
is measured. Determine the best estimate
for
.