Prof: | Robert L. Wolpert |
wolpert@stat.duke.edu | OH: Thu 2-3pm, 211c Old Chem (684-3275) | ||

TA: | Natesh Pillai |
natesh@stat.duke.edu | OH: Tue 3-5pm, 222 Old Chem (684-8840) | ||

Class: | Mon/Wed 2:50-4:05pm | 025 Old Chem | |||

Text: | Sidney Resnick, | A Probability Path | |||

Opt'l: | Patrick Billingsley, |
Probability and Measure (3rd edn); |

Week | Topic | Homework Problems | Due | Sol |
---|---|---|---|---|

I. Foundations of Probability | ||||

Jan 11,18 | Probability spaces: sets, events, and sigma-fields | 1/1,3,4,12,24,39 | Feb 02 | sol |

Jan 23-25 | Probability spaces: Constructing & extending measures | 2/8,11,14,19 | Feb 09 | sol |

Jan 30-Feb 1 | Random variables and their distributions I | 3/2,3,6,15 | Feb 16 | sol |

Feb 6-8 | Random variables and their distributions II | 4/1,2,4,7,8,9,10,11 | Feb 23 | sol |

Feb 13-15 | Independence, sigma-fields & zero-one Laws | 4/13,16,18,19,21,24,28 | Mar 02 | sol |

Feb 20-22 | Integration & expectation I (Lebesgue's theorems) | 5/1,4,7, 16, 17 | Mar 09 | |

Feb 27-Mar 1 | Integration & expectation II (Fubini's theorem) | 5/19,21,24,29,32 | Mar 09 | |

Mar 6-8 | Review and in-class Midterm Exam (Wed Mar 9) | '02, '03, '04a,b, '05 | Results | |

--- Spring Break (Mar 11-19) --- | ||||

II. Convergence of Random Variables &
Distributions | ||||

Mar 20-22 | Convergence concepts: a.s., i.p., L_{p},
L_{oo} |
6/5,7,8,10,13,14,17,30 | ||

Mar 27-29 | Strong & weak laws of large numbers | Martingale Notes | ||

Apr 3-5 | Convergence in distribution & C.L.T. | 7/1,4,8,43,44,46 | ||

Apr 10-12 | Stable limit theorem & ID limits (notes: ps, pdf) | 8/2,3,12; 9/5,6,9,10 | ||

III. Conditional Prob & Expectation | ||||

Apr 17-19 | Radon-Nikodym thm and conditional probability | 10/6,7,8,10,13 | MGs | |

May 5 | Take-home Final Examination (due 7pm). | '02, '03, '04, '05 | Results |

Students are expected to be well-versed in real analysis at the level of
W. Rudin's *Principles of Mathematical Analysis* or M. Reed's
Fundamental Ideas of Analysis--- the topology of R^{n},
convergence in metric spaces (especially uniform convergence of functions on
R^{n}), infinite series, countable and uncountable sets, compactness
and convexity, and so forth. Most students who majored in mathematics will
be familiar with this material; students with less background in math should
consider taking Duke's Math 203, Basic Analysis I. It is also possible to learn the material
by working through standard text, doing most of the problems, preferably in
collaboration with a couple of other students and with a faculty member to
help out now and then.
More advanced mathematical topics from real analysis, including parts of
measure theory, Fourier and functional analysis, are introduced as needed to
support a deep understanding of probability and its applications. Topics of
later interest in statistics (*e.g.*, regular conditional density
functions) are given special attention, while those of lesser statistical
interest (*e.g.*, extreme value theorems) may be omitted.

Some problems and projects may require computation; you are free to use
whatever environmnent you're most comfortable with. Most people find
`R` (some notes are available) or
`Matlab` (a primer is available)
easier to use than compiled languages like `FORTRAN` or `C`.
Homework problems are of the form chapter/problem from the text. Not all of
them will be graded, but they should be turned in for comment; Tuesday
classes will begin with a class solution of one or two of the preceeding
week's problems. Some weeks will have lecture notes added (click on the
"Week" column if it's blue or green). This is syllabus is
*tentative*, last revised , and will almost surely be
superceded- RELOAD your browser for the current version.