
Math 561 Probability I
Lecture Slides
- Lecture 1
- Lecture 2
- Lecture of 02/02
- Lecture of 02/04
Homeworks
- Homework 1: 1.6.2, 1.7.3, 2.1.1, 2.1.9. Due 02/07
- Homework 2: 2.2.1, 2.2.2, 2.2.4, 2.2.5. Due 02/14
- Homework 3: 2.3.15, 2.3.18, 2.4.1, 2.4.2. Due 02/21
- Homework 4: 2.5.1, 2.5.2, 2.5.9, 2.5.10. Due 02/28
- Homework 5: 3.2.6, 3.2.12, 3.2.13, 3.2.14. Due 03/23
- Homework 6: 3.3.13, 3.4.5, 3.4.6, 3.4.12. Due 03/30
- Homework 7: 4.1.8, 4.1.9, 4.1.10, 4.1.11 from the file Materials on Random Walks. Due 04/06
- Homework 8: 4.1.5, 4.1.7, 4.1.8, and this problem Due 04/18
- Homework 9: 4.2.5, 4.2.9, 4.2.10, and this problem Due 04/27
- Homework 10: 4.3.11, 4.4.3, 4.4.6 and 4.4.7 Due 05/02
Class Diary
- Wednesday, 01/19: sigma-field, measurable space, measure, probability measure, measure space, probability space,measures on the real line, Stieltjes function
Read: Section 1.1
- Friday, 01/21:measures on high dimensional Euclidean space, random variables, distributions, distribution functions
Read: Sections 1.1 and 1.2
- Monday, 01/24: Random variables, integration
Read: Sections 1.3 and 1.4
- Wednesday, 01/26: Integration
Read: Section 1.4
- Friday, 01/28: Jensen's inequality and Holder inequality
Read: Section 1.5
- Monday, 01/31: bounded convergence theorem, Fatou's lemma, monotone convergence theorem, expected values
Read: Sections 1.5 and 1.6
- Wednesday, 02/02: Theorem 1.6.8, change of variable formula, \pi-system, \lambda-system, \pi-\lambda theorem, product measures, Fubini's theorem
Read: Sections 1.6 and 1.7
- Friday, 02/04:Independence, sums of independent random variables
Read: Section 2.1
- Monday, 02/07: Kolmogorov's extension theorem, L^2 weak law, polynomial approximation
Read: Sections 2.1.4 and 2.2.1
- Wednesday, 02/09: Triangular arrays, coupon colelctor's problem, occupancy problem, weak law for triangular arrays, weak law of large numbers
Read:Section 2.2., 2.2.3
- Friday, 02/11: weak law of large numbers, St.Petersburg paradox, Borel-Cantelli lemma
Read: Sections 2.2.3, 2.3
- Monday, 02/14: L^2 strong law of large numbers, 2nd Borel-Cantelli lemma,
Read:Section 2.3
- Wednesday, 02/16: head runs, strong law of large numbers, renewal theory,
Read: Sections 2.3 and 2.4
- Friday, 02/18: empirical distribution, The Glivenko-Cantelli theorem, tail events, Kolmogorov's 0-1 law, Kolmogorov's maximal; inequality
Read:Sections 2.4 and 2.5
- Monday, 02/21: Kolmogorov's 3 series theorem, rates of convergence
Read:Section 2.5
- Wednesday, 02/23: Infinite mean
Read: Section 2.5.2
- Friday, 02/25: Weak convergence, characterization of weak convergence,
Read:Sections 3.1, 3.2.1 and 3.2.2
- Monday, 02/28: continuous mapping theorem, Portmanteau theorem, Helly's selection theorem, tightness
Read:Section 3.2.2
- Wednesday, 03/02:tightness, characteristic functions, properties of characteristic functions, the inversion formula
Read: Sections 3.2.2, 3.3.1
- Friday, 03/04: the inversion formula, continuity theorem
Read:Sections 3.3.1 and 3.3.2
- Monday, 03/07: moments and derivatives, Polya's criterion
Read:Sections 3.3.3 and 3.3.4
- Wednesday, 03/09: Centrla limit theorem
Read: Section 3.4.1
- Friday, 03/11: Midterm
- Monday, 03/21: Lindeberg-Feller theorem, the converse of the three series theorem
Read:Section 3.4.2
- Wednesday, 03/23: Infinite variance, rates of convergence
Read:Section 3.4.2, 3.4.4, 2.5
- Friday, 03/25: permutable events, exchangeable sigma-field, Hewitt-Savage 0-1 law
Read:Section 2.5 (or Section 4.1 in the file Materials on Random Walks)
- Monday, 03/28:stopping times and their applications
Read: Section 4.1 in the file Materials on Random Walks.
- Wednesday, 03/30: Wald's equation, simple random walks, Wald's 2nd equation
Read: Section 4.1 in the file Materials on Random Walks.
- Friday, 04/01: Applications of Wald's 2nd equation to random walks, recurrence, transience, simple symmetric random walks on Z^d
Read:Sections 4.1 and 4.2 in the file Materials on Random Walks.
- Monday, 04/04: recurrence and transience, simple symmetric random walk on Z^d
Read:Section 4.2 in the file Materials on Random Walks.
- Wednesday, 04/06: recurrence and transience of general random walks in R^d.
Read:Section 4.2 in the file Materials on Random Walks.
- Friday, 04/08: recurrence and transience of general random walks in R^d (cont)
Read:Section 4.2 in the file Materials on Random Walks.
- Monday, 04/11: Trul3 3d random walks are transient. conditional expectation
Read:Section in the file Materials on Random Walks. Section 4.1 in the book.
- Wednesday, 04/13: Examples and properties of conditional expectations
Read:Sections 4.1.1 and 4.1.2
- Friday, 04/15: Martingales, submartingales, supermartingales, examples, properties
Read:Section 4.2
- Monday, 04/18: upcrossing inequality, martingale convergence theorem
Read:Section 4.2
- Wednesday, 04/20: bounded increments, Doob's decomposition, branching processes
Read:Sections 4.3.1 and 4.3.4.
- Friday, 04/22: branchung processes, Doob's inequality, L^p maximal inequality
Read:Sections 4.3.4 and 4.4
- Monday, 04/25: , L^p convergence theorem, applications to branching processes, uniform integrability and L^1 convergence
Read:Sections 4.4 and 4.6
- Wednesday, 04/27: L^1 convergenc of martingales, dominated convergence theorem for conditional expectations
Read:Section 4.6
- Friday, 04/29: backwards martingale, application to strong law of large numbers, optional stopping theorem
Read:Sections 4.7 and 4.8
- Monday, 05/02: optional stopping theorem
Read:Section 4.8
- Wednesday, 05/04:
Read:Section
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