** 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.

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*

- Chapter 1: Measure Theory and Laws of Large Numbers (plus topics from Lange chapter 2).
- Chapter 2: Stein's Method and Central Limit Theorems (plus topics from Lange chapter 12).
- Chapter 3: Conditinal Expectation and Martingales (plus topics from Lange chapter 10).
- Chapter 4: Bounding Probabilities and Expectation (plus topics from Lange chapter 3).
- Chapter 5: Markov chains (plus topics from Lange chapter 7).
- Poisson processes (Lange Chapter 6).
- Continuous time Markov chains (Lange Chapter 8).
- Branching processes (Lange Chapter 9).
- Chapter 7: Brownian Motion (plus topics from Lange chapter 11).

**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.

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.