Lecture notes (ST 778- Fall 2007)

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• No class August 23
• Lecture 1. lecture 1 (pdf) This lecture is an introduction to set theory (background material for the course), it is not in the textbook. The rest of the lectures will follow closely the text book. August 28.
• Lecture 2 (finishing set theory). Classes of sets (pages 18-22 from Billingsley). HW 1 assigned.
• Lecture 3 Probability measure. Pages 22-27 and page 12 (theorem 1.3). HW 2 assigned (question 1 in HW 2 is extra-credit (optional) 5 extra points out of 60). 9/6/07 and 9/11/07.
• Lecture 4. Existence and extension. Pages 36-40. HW 3 assigned. 9/13/07.
• Lecture 5. Uniqueness of the extension. Pages 41-42. 9/18/07.
• Lecture 6. Completeness and nonmeasureble sets. Independent events. Pages 44-45, 53-55. 9/20/07. HW4 assigned.
• Lecture 7. Independent events and Borel Cantelli lemmas. 53-56 pages. 59-62 pages. 9/25/07.
• Lecture 8. Random variables. 67-73 9/27/07 HW 5 assigned.
• Lecture 9. Expected value and Strong law. Pages 76,77, 80. 85. 10/2/07.
• Lecture 10. Measures. Pages 158-164. 10/4/07. HW 6 assigned.
• Lecture 10. Measures and examples (continuation) (10/9/07). (please download hw6 with the new hints added).

Fall Break. No class October 11.

· Lecture 11. Measures (continuation) and review for the exam. 10/16/07. Homework due. (not homework assigned this week-exam preparation week). MATERIAL FOR EXAM.

Solutions for exam

· Lecture 12. Continuation with measures and examples. HW 7 assigned today. Pages 158-166 (measures). Pages 175-176 (measures on the line). 10/25/07.

· Lecture 13. Measurable MAPPINGS, limits and measurability. Section 13 in the book. 10/30/07. pages 171, 182-184

· Lecture 14. Weak convergence. (section 14 in the book). HW 8 assigned today. 11/1/07 11/6/07. pages 187-189 (page 162)

· Lecture 15. Integration HW9 assigned today. 11/8/07. Section 15. pages 199-204, page 185.

· Lecture 16. Properties of the integral. 11/13/07. pages 206-211.

· Lecture 17. Properties of the integral. 11/15/07. entire section 16. HW10 assigned today, due Nov. 27.

· Lecture 18. Densities, Lebesgue and Riemann integral. Fubini’s theorem 11/20/07. pages 212-217,221-224, 231. It would be helpful if you review some calculus: fundamental theorem of calculus, the change of variable in integrals, fubini theorem, integrals of trigonometric functions.

· Lecture 19. Expected values and moments. Section 21 in the book. 11/27-29/07. HW 11 assigned.

· Lecture 20. Conditional distribution.

· Review for final 12/6/07.

FINAL EXAM (see syllabus for the course, it is DEC 11, 1-4pm) PLACE: Mann Hall #307. You can bring a sheet with formulae (both sides, font 11 points). PREPARATION FOR FINAL (6 questions, level of difficulty similar to hws and midterm). Remember to review criteria for convergence of series of real numbers.

FINAL EXAM

SOLUTIONS TO FINAL EXAM

It would be a good practice to attend all the lectures.

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Last updated August 2, 2007.

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