## STAT 204: Probability for Applications (Fall 2008)

NEW: extra Office Hours (Partha Dey): Monday, Dec 15-th 2-4 pm and Wednesday Dec 17-th 11-12, room 307 Evans

NEW: sample final. Note this year's final is not in the same "write on exam" format -- bring your own paper.

Instructor: David Aldous

Class Time: MWF 1:00 - 2:00 in room 2 Evans.

Teaching Assistant: Partha Dey:

Discussion session Mondays, 3.00 - 4.00, room 340 Evans. Starting September 8.

This is a second course in Probability (prerequisite: an undergraduate course) aimed at graduate students in the Statistics, Biostatistics, Computer Science, Electrical Engineering, Business and Economics Departments who expect their thesis work to involve probability.

In contrast to STAT 205 (which emphasizes rigorous proof techniques) this course will emphasize describing what's known and how to do calculations in a broader range of probability models. Students are encouraged to learn by doing exercises.

The discussion section is optional and will be used (according to student demand) to expand upon lecture material and to work practice problems.

### Texts

Required:
Sheldon Ross and Erol Pekoz A Second Course in Probability. Probabilitybookstore.com
Kenneth Lange Applied Probability. Springer.

A longer book, covering most of the same topics in more depth, and covering more topics, is G. Grimmett and D. Stirzaker Probability and Random Processes

### Approximate Schedule

The Ross-Pekoz book tends to emphasize proofs while the Lange book emphasizes calculations. I will mostly follow the order of Chapters (below) in Ross-Pekoz while sometimes substituting material from Lange.
1. Chapter 1: Measure Theory and Laws of Large Numbers (plus topics from Lange chapter 2).
2. Chapter 2: Stein's Method and Central Limit Theorems (plus topics from Lange chapter 12).
3. Chapter 3: Conditinal Expectation and Martingales (plus topics from Lange chapter 10).
4. Chapter 4: Bounding Probabilities and Expectation (plus topics from Lange chapter 3).
5. Chapter 5: Markov chains (plus topics from Lange chapter 7).
6. Poisson processes (Lange Chapter 6).
7. Continuous time Markov chains (Lange Chapter 8).
8. Branching processes (Lange Chapter 9).
9. Chapter 7: Brownian Motion (plus topics from Lange chapter 11).

### Homework

Weekly homework posted here by Fridays, due in class each Wednesday (first homework due Wed Sept 10).

50% homework, 50% final.

### Final exam

The campus schedule of classes may give the impression there's no final exam -- but there is. Wednesday, December 17, 5.00 - 8:00 pm, room 334 Evans.

### Handwriting Rule

You can bring to the final exam anything in your own handwriting but nothing else. So you can bring your notes from class, your homework, summaries of the course material you have made - provided these are literally in your own handwriting without electronic intermediation.

### Miscellaneous

Here is a long list of practice final problems compiled several years ago by Shankar Bhamidi. But these are aimed at the Lange book.

### Office Hours

David Aldous (aldous@stat) Tuesdays 1.30 - 3.30 in 351 Evans

Partha Dey (partha@stat) Mondays 4.00 - 5.00; Wednesday 11.00 - 12.00; Wednesday 4.00 - 5.00 (all in 307 Evans).

If you email us please put "STAT 204" in subject.

#### Topic each class.

W 8/27: Style of course. Undergrad final exam (as review).
F 8/29: Review of Expectation.

W 9/3: Measure theory: convergence of RVs and expectations.
F 9/5: Probability distributions.

M 9/8: Coupling.
W 9/10: Stein-Chen Poisson approximation: coupling version.
F 9/12 : examples (independent; coupon collector; large spacings).

M 9/15 : neighborhood version: example.
W 9/17 : conditional expectation.
F 9/19 : martingales -- definition and examples.

M 9/22 : stopping times, optional stopping theorem, Wald's equation, play red.
W 9/24 : fair game principle, poker tournament, Wright-Fisher fixation, patterns in IID sequences.
F 9/26 : boundary crossing; Azuma-Hoeffding inequality.

M 9/29 : Method of bounded differences, empty boxes, chromatic number of random graph.
W 10/1 : submartingales, convergence theorems, maximal inequalities.
F 10/3 : Inequalities (Markov, Jensen, Cauchy-S); use of change of density in Normal tail bounds.

M 10/6 : Second moment inequality, conditional expectation inequality.
W 10/8 : Expectations of maxima.
F 10/10 : Inclusion-Exclusion.

M 10/13 : the MM algorithm.
W 10/15 : Markov chains, transition matrix, etc
F 10/17 : stationary distributions.

M 10/20 : examples of stationary distributions.
W 10/22 : classification of states; statements of limit theorems.
F 10/24 : Coupling proof of convergence theorem. MCMC.

M 10/27 : Poisson distribution; construction and definition of abstract Poisson process.
W 10/29 : 5 general properties of abstract Poisson processes.
F 10/31 : Examples (traffic, postcards). Campbell formula.

M 11/3 : 1-dimensional Poisson processes.
W 11/5 : Continuous-time Markov chains.
F 11/7 : Simple examples.

M 11/10 : Kendall birth-death-immigration process.
W 11/12 : Galton-Watson branching processes: moments and extinction probabilities.
F 11/14 : Just-supercritical and conditioned GWBP. Lange's HIV example.

M 11/17 : Multitype continuous-time BPs. *** no lab today ****
W 11/19 Lab session in class slot
F 11/21 **** no class today ****

M 11/24 : Brownian motion as limit of SRW; definition of 1-dim diffusion; change of variables formula.
W 11/26 : 2 examples of diffusions as limits of discrete processes: Ehrehfest urn odel and Wright-Fisher diffusion.

M 12/1 : Formula for first exit place; d-dimensional Brownian Motion.
W 12/3 : Formulas for stationary density and mean exit time; examples.
F 12/5 : Martingales associated with BM; geometric BM; Kelly diffusion.

M 12/8 : Review: cat and mouse; cover N-cycle; nearest neighbors.
W 12/10 : Review; bus schedule, umbrella, Buffon needle.