## M564 - Probability

Section: 17882
Class Day and Times: MWF 11:15-12:05
Class Location: BH 246
WEB Page: http://elizabethhousworth.com/Spring2012/M564
Text: Probability and Measure by Patrick Billingsley
Instructor: Elizabeth Housworth
Office and Office Hours: 371 Rawles Hall, 1-4 pm Sundays and by appointment

Important Dates: Last day for Automatic Withdrawal: Wednesday, March 7. Exam 1: Monday, February 13. Exam 2: Monday, March 26. Final Exam: 5-7 p.m., Monday, April 30.

Religious Holiday Policy: If you will miss class, especially a class during which there will be an exam or other required work, for a religious holiday, you must inform me during the first two weeks of the semester.

Cheating: It is not possible to cheat on homeworks. All exams must be your own individual work. All suspected cases of cheating will be handled in accordance with University procedures found at http://dsa.indiana.edu/ethics.html . If you do cheat, you will receive an F* in the course - the star informs the registrar that the F is due to cheating. Additional sanctions may be imposed by the Dean.

Homework: There will be 10 homework assignments throughout the term. See the schedule below. A star (*) on a problem indicates that a hint to the problem appears in the back of the text. At least one assignment will be dropped from the calculation for your homework average.

Grades: Grades are calculated as 40% Homework, 15% Exam 1, 15% Exam 2, 25% Final Exam, 5% Attendance and Participation. In the rare event that a student who has failed to turn in their homeworks consistently on time performs extraordinarily well (as in nearly perfectly) on the in-class exams, an alternative weighting will be used that discounts the homework contribution to the course grade.

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### Tentative Syllabus

Date Lecture Topics Section
Monday, January 9 Poisson point processes (through Thm 23.1) Section 23
Wednesday, January 11 Poisson point processes (through condition 3) Section 23
Friday, January 13 Poisson point processes (condition 4 and stochastic processes) Section 23
Homework 1: (due Friday, Jan. 20) 23.3*, 23.4*, 23.9*, 23.10 (read 23.11 for an application, but it is not assigned to turn in)
Monday, January 16 Martin Luther King Jr. Holiday --
Wednesday, January 18 Weak convergence (through Thm 25.4) Section 25
Friday, January 20 Fundamental theorems Section 25
Monday, January 23 Helly's theorem and integration to the limit review Section 25
Wednesday, January 25 Characteristic functions Section 26
Friday, January 27 Inversion and uniqueness Section 26
Homework 2: (due Monday, January 30) 25.4, 25.17, 25.18, 25.20*
Monday, January 30 Continuity theorem Section 26
Wednesday, February 1 Central Limit Theorem (easy case) Section 27
Friday, February 3 Lindenberg and Lyapounov Theorems Section 27
Homework 3: (due Monday, February 6) 26.1* 26.20, 26.21, 27.1
Monday, February 6 Multivariate analogs - limit theorems Section 29
Wednesday, February 8 Multivariate analogs - Central Limit Theorem Section 29
Friday, February 10 Selected topics on derivatives on the line Section 31

Monday, February 13 Exam 1 (Covering through the easy version of the Central Limit Theorem)
Wednesday, February 15 Signed measures Section 32
Friday, February 17 Radon-Nikodym Theorem Section 32
Homework 4: (due Monday, February 20) 27.4, 27.5, 27.10, 29.2
Monday, February 20 Conditional probabilities Section 33
Wednesday, February 22 Conditional probabilities Section 33
Friday, February 24 Conditional probabilities Section 33
Homework 5: (due Monday, February 27) 29.12, 29.13, 31.1, 31.2
Monday, February 27 Conditional expectation Section 34
Wednesday, February 29 Conditional expectation Section 34
Friday, March 2 Martingales: definitions and examples Section 35
Homework 6: (due Monday, March 5) 32.4, 33.3*, 33.5(a), 33.8
Monday, March 5 Submartingales and gambling Section 35
Wednesday, March 7 Stopping times Section 35
Friday, March 9 Convergence theorems Section 35
March 10-18 Spring Break
Homework 7: (due Monday, March 19) 34.3*, 34.4*, 34.6, 34.11*
Monday, March 19 Applications of martingales Section 35
Wednesday, March 21 Applications of martingales Section 35
Friday, March 23 Stochastic processes and review Section 36

Monday, March 26 Exam 2 (Covering through martingales)
Wednesday, March 28 Kolmogorov's Existence Theorem Section 36
Friday, March 30 Inadequacies Section 36
Homework 8: (due Monday, April 2) 35.2, 35.4*, 35.5, 35.6
Monday, April 2 Brownian motion Section 37
Wednesday, April 4 Continuity of paths Section 37
Friday, April 6 Measurability, irregularity, scaling properties Section 37
Homework 9: (due Friday, April 13) 36.2, 36.7*, 37.1*, 37.2
Monday, April 9 Set of zeros Section 37
Wednesday, April 11 Strong Markov property Section 37
Friday, April 13 Reflection principle Section 37
Monday, April 16 Invariance Section 37
Wednesday, April 18 Harmonic functions and Brownian motion (supplemental) Section 37
Friday, April 20 Harmonic functions and Brownian motion (supplemental) Section 37
Homework 10: (due Monday, April 23) 37.11, 37.12, 37.13, 37.14*
Monday, April 23 Supplemental topics on Brownian motion Section 37
Wednesday, April 25 Supplemental topics on Brownian motion Section 37
Friday, April 27 Summary and evaluations
Final Exam: Monday, April 30 5-7 pm in BH 246