Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 1 Solutions
 Create a list of all the stochastic processes you can think of that might occur in the real world (not just examples from the textbook). Be creative!

We defined a field to be a collection of sets that is closed
under complementation and finite unions. Show that such a
collection is also closed under finite intersections.
Given the field we know that for , we have and . Generalizing, when , then . Furthermore, , as must be the complementary event,

Using the axioms of probability, prove the following properties
of probability:

and .
but also (by additivity), so .

.

If then . So


This is true for events: . Proof by induction: Assume true for :


Suppose
. Prove the following properties of
conditional probability:

.
. But since , the result follows.

.

For
with
for
,
, since the sets are disjoint.



Since , . Then

.
Since , . Then

.

Prove the law of total probability.
Let be a partition of . Note that . Note also that .

Prove Bayes rule

Suppose
and
are independent events. Show that
and
are also independent.
By independence, .