Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 2 Solutions

Suppose
is a r.v. with c.d.f.
. Prove the following:

is nondecreasing.
Let .

.

.

is right continuous.
Let for . Note that this is a nested sequence, . We have

if
.
.

.

is nondecreasing.

Show that the following are valid p.m.f.s:

Binomial:
if
.
Need to show that . Use the binomial theorem:

Poisson:
for
.
Need to show that .

Binomial:
if
.

Find the mean and variance of
when
is

;
Let , thereforeNow . So, .

Binomial
;
Binomial p.m.f is given by 
Poisson
;
Poisson : 
Exponential
;
Exponential : .

;

Suuppose that
and
are jointly continuous. Show that

Suppose that
and
are jointly Gaussian with
parameters
. Show that
.
In this case we have, 
Suppose
, and define
. Are
and
uncorreleated? Are
and
independent?
Find the pdf of
. Are
and
jointly continuous?
Note that and . Then
X and Y are jointly continuous: Look at the joint CDF: