Probability Spaces, Measures, and Integration

MATH 6221

Spring 2013

Skiles 169

T / Th, 4:35 - 5:55 pm


Will Perkins

Office: Skiles 017

  • Sections: 1.2 - 1.6, 5.6
  • Notes: Probability Spaces, Measures and Integration, Independence. Appendix A.4 - A.6 of Durrett.
  • Homework Problems: HW 1: 1, 2, 6, 7, 8, 10, 11, 12, 13
  • Definitions and Theorems:
    1. Probability Space
    2. Outcome
    3. Event
    4. Sigma Field
    5. (Probability) Measure
    6. Product Space
    7. Borel Sigma Field
    8. Sigma field generated by a collection of sets
    9. Measurable Function
    10. Simple function
    11. Definition of an Integral
    12. Jensen's Inequality
    13. Holder's Inequality
    14. Cauchy-Schwartz Inequality
    15. Bounded Convergence Theorem
    16. Monotone Convergence Theorem
    17. Fatou's Lemma
    18. Dominated Convergence Theorem
    19. Fubini's Theorem
    20. Independent Events
    21. Conditional Probability
    22. Bayes' Rule
  • Sample Questions:
    1. Give an example of a function from a space Omega -> R that is measurable under one sigma field but not measurable under another.
    2. When can a limit be taken inside an integral? Give an example when it cannot.
    3. Consider flipping a fair coin 10 times. Let X be the number of heads. Describe the sigma field generated by X.
    4. Outline a proof of why integration is linear. (I.e. \int (f+g) = \int f + \int g )
    5. Let f be the indicator function of the rational numbers. Prove that f is not Riemann integral, and calculate its Lebesgue integral.
    6. Describe a probability space for the following experiment: you deal a hand of 5 cards from a standard 52 card deck, shuffle, and repeat 10 times.
    7. Flip a coin 10 times. Let A be the event that exactly 5 flips were heads, and let B be the event that the last flip was a head. Are A and B independent? Let C be the event that the first flip was a tail. Describe the dependencies of the three events.
    8. If we want to do a statistical survey (i.e. a political poll) what is the implied probability space?
    9. Give an example of a probability space whose sigma field contains uncountably many events.
    10. Give an example of a measurable function which is not integrable.
    11. Give an example of a probability measure on the non-negative integers, and integrate the function f(x) = x^2 with respect to that measure.