Rice University
This course covers the basic concepts of probability theory
and random processes.
Targeted at first year graduate students it introduces concepts
at an appropriately rigorous level and discusses applications through examples
and homework, such as to Digital Communication Systems. The syllabus covers
elementary probability theory, random variables, limiting theorems such as
the Law of Large Numbers, the Central Limit Theorem, and Martingales,
as well as Gaussian, Markovian and Renewal Processes.
Classes
| Covered material | Reading: Stark&Woods (2002) | |
| August 27 | Orientation, history | |
| August 29 | Probability space: basics | Review combinatorics: 24-31 |
| September 3 | Probability space: discrete, continuous | 1-24 |
| September 5 | Random variable, CDF, pdf | 58-68 |
| September 10 | Conditional Probability, Bayes, Independence | 68-80 |
| September 12 | Functions of one r.v., expectation | 116-134, 169-175 |
| September 15 | Moments, Independent experiments | pp 192-196, 32-44 |
| September 17 | Joint distributions, Marginals, Independent r.v. | pp 88-99 |
| Up to here: Material for Quiz | ||
| September 22 | Functions of two r.v., Sums and Products | pp 134-152 |
| September 24 | Covariance, Stieltjes integral | |
| September 26 | QUIZ | |
| September 29 | NO CLASS, moved to Oct 3 | |
| October 1 | generalized pdf (Dirac), conditional CDF | 75-80, 80-88 |
| October 3 | conditional pdf, E[Y|X]: rules | 103-108 |
| October 6 | E[Y|X]: several variables | 183-192 |
| October 8 | MMSE, E[Y|X] as a projection, Gaussian estimation | 552-561 |
| October 13 | RECESS, moved to Oct 17 | |
| October 15 | Characteristic function | 216-225 |
| October 17 | Joint char fct, joint Gaussian | 277- 280, 281 (also: 269-277) |
| October 20 | Inequalities, Convergence of functions | pp 205-210, 375-376 |
| October 22 | Convergence of random variables | pp 376-382 |
| October 27 | Limit theorems | pp 383-387, 225-230, 214-216 |
| October 29 | Limit theorems | |
| Up to here: Material for TEST 1 | ||
| November 5 | Random Processes: Basics | pp 401-407 |
| November 7 | Consistency, Stationarity | |
| November 12 | Renewal Processes: Basics | pp 408-416 |
| November 14 | Poisson Process | |
| November 19 | Consistency: Gaussian Processes | pp 418-421 |
| November 21 | Consistency: Markov (Chapman-Kolmogorov) | pp 421-430 |
| November 24 | Spectral density | 348-354 |
| November 26 | Cross-correlation and -spectrum, Mean square continuity | 348-354, pp 487-490 |
| November 28 | Thanksgiving | |
| From Test 1 to here: Material for Test 2 | ||
| December 1 | Mean square calculus | pp 487-497 |
| December 3 | More of Spectral density, White Noise, Karhunen-Loewe |