Probability Spaces, Measures, and Integration
T / Th, 4:35 - 5:55 pm
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:
- Probability Space
- Sigma Field
- (Probability) Measure
- Product Space
- Borel Sigma Field
- Sigma field generated by a collection of sets
- Measurable Function
- Simple function
- Definition of an Integral
- Jensen's Inequality
- Holder's Inequality
- Cauchy-Schwartz Inequality
- Bounded Convergence Theorem
- Monotone Convergence Theorem
- Fatou's Lemma
- Dominated Convergence Theorem
- Fubini's Theorem
- Independent Events
- Conditional Probability
- Bayes' Rule
- Sample Questions:
- Give an example of a function from a space Omega -> R that is measurable under one sigma field but not measurable under another.
- When can a limit be taken inside an integral? Give an example when it cannot.
- Consider flipping a fair coin 10 times. Let X be the number of heads. Describe the sigma field generated by X.
- Outline a proof of why integration is linear. (I.e. \int (f+g) = \int f + \int g )
- Let f be the indicator function of the rational numbers. Prove that f is not Riemann integral, and calculate its Lebesgue integral.
- 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.
- 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.
- If we want to do a statistical survey (i.e. a political poll) what is the implied probability space?
- Give an example of a probability space whose sigma field contains uncountably many events.
- Give an example of a measurable function which is not integrable.
- 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.