Homework (due Fridays in class):
You are encouraged to work together on the homework, but I ask that
you write up your own solutions and turn them in separately.
HW #1, due Friday, January 18, in class: the info sheet.

HW #2: to be announced in class.

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Topics:

This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics: (1) the basic limit theorems (the law of large numbers, the central limit theorem and the large deviation principle); (2) martingales and their applications. If time allows, we will give a brief introduction to Brownian motion. The prerequisite for Math 561 is Math 540.

### Why study probability?

Two main reasons: uncertainty and complexity.
Uncertainty is all around us and is usefully modeled as randomness: it appears in call centers,
electronic circuits, quantum mechanics, medical treatment,
epidemics, financial investments, insurance, games (both sports and
gambling), online search engines, for starters.
Probability is a good way of quantifying and discussing what we know
about uncertain things, and then making decisions or estimating
outcomes.
Some things are too complex to be analyzed exactly (like weather, the
brain, social science), and probability is a
useful way of reducing the complexity and providing approximations.
And the reason I study probability: statistical mechanics,
which combines both the uncertainty of quantum mechanics, and the
complexity of zillions of particles interacting.