## M563 - Probability

Section: 4448
Class Day and Times: MWF 11:15-12:05
Class Location: SW 221
WEB Page: http://elizabethhousworth.com/Fall2011/M563
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, October 26. Exam 1: Monday, October 3. Exam 2: Monday, November 7. Final Exam: 10:15 a.m.-12:15 p.m., Monday, December 12.

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 11 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. 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.

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### Tentative Syllabus

Date Lecture Topics Section
Monday, August 29 length, dyadic intervals, Rademacher functions, step functions, Chebyshev's inequality for step functions, weak law of large number. Section 1
Wednesday, August 31 Negligible sets, strong law of large numbers, normal numbers, length Section 1
Friday, September 2 Sets, fields, sigma-fields, probability measure Section 2
Homework 1: (due Wednesday, Sept. 7) 1.1*, 1.2, 1.3*, 1.8, 2.1
Monday, September 5 Labor Day Holiday --
Wednesday, September 7 Lebesgue meaure, cylinder sets, outer measure Sections 2-3
Friday, September 9 Existence and uniqueness Section 3
Homework 2: (due Monday, September 12) 2.3*, 2.5*, 2.11*, 2.13, 2.20
Monday, September 12 conditional probability, limit sets, independence Section 4
Wednesday, September 14 subfields, Borel-Cantelli lemmas Section 5
Friday, September 16 Zero-One law, simple random variables Sections 4-5
Homework (due Monday, September 19) 3.5, 3.11, 4.2(a), 4.6, 4.11
Monday, September 19 convergence, independence Section 5
Wednesday, September 21 Expected value Section 5
Friday, September 23 Law of large numbers Section 6
Homework 4: (due Monday, September 26) 5.3 (assume X is simple), 5.5*, 5.12 (assume X is discrete but not necessarily simple), 6.4, 6.6
Monday, September 26 gambler's ruin, gambling strategies Section 7
Wednesday, September 28 gambling strategies Section 7
Friday, September 30 introduction to Markov chains Section 8

Monday, October 3 Exam 1 (Covering sections 1-6)
Wednesday, October 5 transience and persistence Section 8
Friday, October 7 stationary distributions Section 8
Homework 5: (due Monday, October 10) 7.2, 7.4, 7.7*, 8.2, 8.3
Monday, October 10 moment generating functions, large deviations Section 9
Wednesday, October 12 Law of the Iterated Logarithm Section 9
Friday, October 14 general measures Section 10
Homework 6: (due Monday, October 17) 8.15, 8.19*, 8.24*, 8.25*, 9.4
Monday, October 17 outer measure Section 11
Wednesday, October 19 measure in Euclidean space Section 12
Friday, October 21 measurable functions Section 13
Homework 7: (due Monday, October 24) 10.4, 11.2, 12.12, 13.1, 13.8
Monday, October 24 distribution functions Section 14
Wednesday, October 26 weak convergence Section 14
Friday, October 28 integration Section 15
Homework 8: (due Monday, October 31) 14.1, 14.2, 14.3*, 14.5*, 15.1*
Monday, October 31 properties of integration: monotonicity, linearity, limits Section 16
Wednesday, November 2 densities and changing variables Section 16
Friday, November 4 uniform integrability Section 16

Monday, November 7 Exam 2 (Covering sections 1-14)
Wednesday, November 9 integration with respect to Lebesgue measure Section 17
Friday, November 11 product measure and Fubini's Theorem Section 18
Homework 9: (due Monday, November 14) 16.7, 16.8*, 16.9*, 17.4, 17.5
Monday, November 14 integration by parts Section 18
Wednesday, November 16 random variables and distributions Section 20
Friday, November 18 independence and sequences Section 20
Homework 10: (due Monday, November 21) 18.4, 18.7, 18.14, 20.5, 20.6
Monday, November 21 convolution and convergence in probability Section 20
Thanksgiving Break
Monday, November 28 expected value, moment generating functions Section 21
Wednesday, November 30 sum of independent random variables, strong law, weak law, moment generating functions, 0-1 law. Section 22
Friday, December 2 maximal inequalities, convergence Section 22
Homework 11: (due Monday, December 5) 20.20*, 21.7, 21.10*, 21.21*, 22.8*
Monday, December 5 exponential distribution and Poisson processes Section 23
Wednesday, December 7 Poisson processes Section 23
Friday, December 9 Summary and evaluations
Final Exam: Monday, December 12 at 10:15 am in SW 221 (Covering sections 1-22)