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Homework 1


G&S, Chapter 1


  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 $P(B) > 0$ . 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 January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: This work is licensed under a Creative Commons License Creative Commons License