Utah State University
ECE 6010
Stochastic Processes
Homework # 1
Due Friday September 9.
Reading G&S, Chapter 1
Exercises
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 copmlementation and finite
unions. Show that such a collection is also closed under finite intersections.
3. Using the axioms of probability, prove the followingproperties of probability:
(a) P(Ac) = 1 −P(A)
(b) P(∅) = 0
(c) A⊂B⇒P(A)≤P(B)
(d) P(A∪B) = P(A) + P(B)−P(AB)
(e) A1, A2,...∈ F ⇒ P(∪∞
i=1)≤P∞
i=1 P(Ai)
4. Suppose P(B)>0. Prove the following properties of conditional probability:
(a) P(A|B)≥0.
(b) P(Ω|B) = 1
(c) For A1, A2,...∈ F with AiAj=∅for i6=j,P(∪∞
i=1Ai|B) = P∞
i=1 P(Ai|B)
(d) AB =∅ ⇒ P(A|B) = 0.
(e) P(B|B) = 1
(f) A⊂B⇒P(A|B)≥P(A)
(g) B⊂A⇒P(A|B) = 1.
5. Prove the law of total probability.
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