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# Homework Solutions

## Utah State University ECE 6010 Stochastic Processes Homework # 1 Solutions

1. 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!
2. 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,

Hence, by DeMorgan's theorem,

3. Using the axioms of probability, prove the following properties of probability:
1. and .

but also (by additivity), so .

2. .

3. If then . So

4.  (*)

Now

so

and so

Substituting these in (**) gives the answer.

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

and show that it holds for :

But the set describes a single set, which we will call . We then have . But by the inductive hypothesis, we have a bound on :

so that

4. Suppose . Prove the following properties of conditional probability:
1. .

. But since , the result follows.

2. .

3. For with for ,

, since the sets are disjoint.

4. Since , . Then

since .

5. .

Since , . Then

5. Prove the law of total probability.

Let be a partition of . Note that . Note also that .

6. Prove Bayes rule

7. Suppose and are independent events. Show that and are also independent.

By independence, .

so

which means and are independent.