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Probability Theory


This assignment is designed to:

  • provide practice with some essential probability theory
  • help you become more fluent with the terminology and the techniques
  • help you grow in confidence in your mathematical abilities


This is a mathematical homework assignment. Show your work. Be clear and concise. Type your assignment.

I strongly recommend you work through these exercises as soon as possible for their instructional content. If you have general questions about the assignment, please post on the Google Group. Finish early, and earn the early bonus.


Question 1: Warming up

[10 points: 2 for the first, 3 each for the second and thirs parts]

Let's begin with the same sample space $\Omega$ we have been using in class: Let $\Omega$ be the set of all outcomes defined by three successive coin flips. Assume that the coin is fair and that the distribution over all singleton events (events containing one outcome) is uniform. Let the event $D$ be the set of all outcomes in which the second flip is a head. Let the event $E$ be the set of all outcomes in which teh final flip is a tail.

  1. What are $P(D)$, $P(E)$, and $P(D \cap E)$ ?
  2. What is $P(D|E)$ ?
  3. What is $P(E|D)$ ?

Question 2: Joint Probability

[10 points]

(Adapted from: Manning & Schuetze, p. 59, exercise 2.3)

Compute the probability of the event 'A period occurs after a three-letter word, and this period indicates an abbreviation (not an end-of-sentence marker)'.

  • Let is-abbreviation denote the event “this period indicates an abbreviation”, and let three-letter-word denote “a period occurs after a three letter word”
  • Assume the following probabilities:
    • $P($is-abbreviation$ | $three-letter-word$) = 0.8$
    • $P($three-letter-word$) = 0.0003$

Question 3: Useful theorems in probability theory

[60 points; 20 per sub‐problem]

(Adapted from: Manning & Schuetze, p. 59, exercise 2.1)

Use the Set Theory Identities and Axioms of Probability Theory to prove each of the following five statements. Develop your proof first in terms of sets and then translate into probabilities; use set theoretic operations on sets and arithmetic operators on probabilities. Be sure to apply good proof technique: justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification, you must first satisfy the conditions / pre-requisites of the axiom.

  1. $P(B - A) = P(B) - P(A \cap B)$
  2. $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule)
    • Hint: refer to part result #1 as a part of your proof of part #2
  3. $P(\neg A) = 1 - P(A)$
    • Hint: refer to part result #1 as a part of your proof of part #4

Question 4: Factoring Joint Distributions

[10 points]

  1. How many possible ways can you completely factor an arbitrary joint probability distribution on six events $P(X_1, X_2, X_3, X_4, X_5, X_6)$
  2. Given the same joint probability distribution $P(X_1, X_2, X_3, X_4, X_5, X_6)$, apply the chain rule to completely factor this distribution in one way.

Question 5: Conditional Probability

[10 points]

Apply the definition of conditional probability to prove Bayes’ Law. Use the same standard of proof as in problem #1 above.


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cs-401r/assignment-1.1410185500.txt.gz · Last modified: 2014/09/08 08:11 by ringger
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