Stat 9300-9310 / Math 6480-6490 - graduate level introduction to probability theory. GENERAL DESCRIPTION: Graduate level means ``with proofs, based on theory of Lebesgue mesaure and integration''. It does not mean ``without intuition'' nor ``purely formal''. Compare to Stat 5100/4300. There they use Ross. He does not even prove our first result (Strong Law of Large Numbers) except in finite variance case. Furthermore, even if you don't care about proofs, he does not even state many (most!) critical results on which classical statistical tests are based, e.g., LIL, large deviations, any theorem or criterion for Poisson approximation to be valid, etc. HOW TO DECIDE IF YOU HAVE THE PRE-REQUISITES FOR THIS COURSE: Students sometimes need some help deciding whether this is the right course for them. If you are in doubt, have a look at the textual materials and the first homework (already posted) and see whether you can envision being comfortable with that material within a few weeks. In general, facility with writing mathematical proofs at the level of math 3600-3610 is going to be much more important than any specific mathematical or statistical knowledge. Mathematical analysis at the level of Penn's Math 5080-5090 is recommended and if you've had measure theory you'll be glad of it, but many students have not had this and in principle an undergraduate level analysis course such as Penn's Math 3600-3610 should be sufficient, PROVIDED YOU LEARNED IT WELL. Most students have had undergraduate probability but those who have not usually do fine. Students who have not learned analysis well at the basic level (Penn 3600-3610) usually struggle mightily. ALTERNATIVE COURSE: Math 5460 (Applied probability modeling) Also provides rigorous, measure-based probability. Emphasis, however, is on probability modeling. Proofs are provided but are not the focus of the course. Rather, we focus on how to construct probability models, how to understand fundamental constructions such as product spaces, weak limits and point processes. We also focus on achieving precise understanding of classical results such as laws of large numbers, Borel-Cantelli, Martingale results and limit theorems (laws of large numbers, central limit theorems, Poisson limit theorems, arcsine laws, etc.). Students will still need a prior course in real analysis but do not need it at the level of 5080-5090. The course is meant to be of particular value to students with existing applied math interests. The main drawback is that, as of now, we can offer only one semester. This covers roughly the first 60% of Math 6480-6490, through martingale theory.