This course covers the fundamentals of Probability Measure, including topics such as Set Theory, Fields, Sigma Fields, Measurable Spaces, Probability Measure, Conditional Probability, Random Variables, and Cumulative Distribution Function. Students will learn about limit supremum and limit infimum, constructing fields, extending probability measures, and the properties of set functions. The teaching method involves theoretical explanations, examples, and theorems to deepen understanding. This course is designed for individuals interested in advancing their knowledge of probability theory and measure theory.
Overview
Syllabus
Probability Measure: 1. Set Theory.
Limit Supremum and Limit Infimum of a Sequence of Real Numbers.
Limit Supremum and Limit Infimum of Sets (part 1 of 2).
Limit Supremum and Limit Infimum of Sets (part 2 of 2).
2 Examples with limsup and liminf.
Probability Measure: 2. Fields.
How to Construct the Smallest Field Containing Sets A1,..., An.
Probability Measure: 3. Sigma Fields.
Probability Measure: 4. Measurable Spaces.
Set Functions on Measurable Spaces.
Properties of Set Functions.
Continuity of a Set Function.
A subset (Vitali set) of the Reals that is not Lebesgue measurable.
Probability Measure: 5. Probability Measure.
Extension of a probability measure from a field to a slightly larger class of sets..
Extension of a probability measure to all subsets of omega.
Outer Measure.
A probability measure on a field, F, can be extended to a probability measure on sigma(F).
Complete Measure.
Example of a completion of a measure space.
Monotone Class Theorem.
Caratheodory Extension Theorem.
1st and 2nd Borel Cantelli Lemmas.
Erdos-Renyi Lemma: Extension of the 2nd Borel-Cantelli Lemma.
Approximation Theorem (Measure Theory).
Probability Measure: 6. Conditional Probability.
Theorem of Total Probability.
Probability Measure: 7. Independence.
Show that R & Theta are Independent in Polar Coordinates.
Probability Measure: 8. Random Variable.
Probability Measure: 9. Functions of Random Variables / Vectors.
Probability Measure: 10 Cumulative Distribution Function.
Riemann Stieltjes Integration for Statisticians.
Example where both the Approximation theorem and Caratheodory Extension Theorem Fail.
Taught by
statisticsmatt
