Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 3 Solutions

Show that
.

Suppose
. Use the ch.f. of
to find
an expression for
,
.

Suppose
and
are the indicator functions of events
and
, respectively. Find
, and show that
and
are independent if and only if
.
are independentindependent

Suppose
is a ch.f. Show that
is also a
ch.f.

Suppose
and
are jointly Gaussian. Use ch.f.s to show
that
.
Now,

Suppose
and
are jointly continuous. (a) Show that
and thus that
Now,
(b) Suppose . Show that

Suppose
and
are independent continuous r.v.s with
c.d.f.s
and
, respectively. Suppose further that
for all
. Show that
(because independent)

Prove Jensen's inequality for the case of simplefunction r.v.'s
First, we prove that the convexity idea generalizes to multiple points. For a convex function we know that
(***)
We'll do it for three points, from which the induction to points should be straightforward. Let . Consider
(*)
(**)
Now to Jensen's inequality. It, too, is proved by induction. We will demonstrate explicitly the first couple of steps. Suppose (a simple function involving a single set . Then takes on two values: , with probability and 0 , with probability . Then
Now consider a simple function involving two disjoint sets:

Prove the Schwartz inequality.
Consider the quantity , which is for all values of the real constant . Expanding, we have
(*)