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Homework Assignments

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Homework 1

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 following properties of probability:
    1. $P(A^c) = 1-P(A)$
    2. $P(\emptyset) = 0$
    3. $A \subset B \Rightarrow P(A) \leq P(B)$
    4. $P(A \cup B) = P(A) + P(B) - P(AB)$
    5. $A_1,A_2,\ldots \in \Fc \Rightarrow P(\cup_{i=1}^\infty) \leq \sum_{i=1}^\infty P(A_i)$
  4. Suppose $P(B) > 0$ 0$" align="middle" border="0" height="37" width="84" />. Prove the following properties of conditional probability:
    1. $P(A\vert B) \geq 0$.
    2. $P(\Omega\vert B) = 1$
    3. For $A_1,A_2,\ldots \in \Fc$ with $A_i A_j = \emptyset$ for $i \neq j$, $P(\cup_{i=1}^\infty A_i\vert B) = \sum_{i=1}^\infty P(A_i\vert B) $

    4. $AB=\emptyset \Rightarrow P(A\vert B) = 0.$
    5. $P(B\vert B) = 1$
    6. $A \subset B \Rightarrow P(A\vert B) \geq P(A)$
    7. $B \subset A \Rightarrow P(A\vert B) = 1$.

  5. Prove the law of total probability.
  6. Prove Bayes rule
  7. Suppose $A$ and $B$ are independent events. Show that $A$ and $B^c$ are also independent.
Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, June 13). Homework Assignments. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/hw1.htm. This work is licensed under a Creative Commons License. Creative Commons License
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