| Lecture 1 | Introduction,
Probability spaces, properties of probability measures,
conditional probability, statistical independence,
conditional independence.
|
| Lecture 2 | Repeated trials,
random variables, CDFs, PMFs, PDFs, mixed and singular random
variables, Multiple random variables, joint
distributions, stochastic processes.
|
| Lecture 3 | The Bernoulli process, expected values,
moments, sums of random variables, conditional expectation.
|
| Lecture 4 | The coupon collector problem, Markov's
inequality, Chebyshev's inequality, Chernov Bounds, moment generating
functions. |
| Lecture 5 | Chernov bound examples, Moment generating
funcitons and sums of independent random variables, log-moment
generating functions. |
| Lecture 6 | Convergence of random variables: mean-squared convergence, convergence in
probability, almost sure convergence,
laws of large numbers. |
| Lecture 7 | The strong law of large numbers, convergence
in distribution, characteristic functions, the central limit theorem. |
| Lecture 8 | Counting Processes and the Poisson Process. |
| Lecture 9 | Splitting and combining Poisson Processes,
Markov Property, Markov Chains: transistion matrices/graphs. |
| Lecture 10 | Markov Chains: first-step analysis,
state classifications, stationary distributions. |
| Lecture 11 | Markov Chains: Stationary distributions,
Balance equations; Introduction to Gaussian Random Vectors. |
| MID-TERM EXAM |
| Lecture 12 | More on Gaussian random vectors. |
| Lecture 13 | Conditioning and Gaussian random vectors;
introduction to Gaussian Processes. |
| Lecture 14 | Stationary processes, Properties of
covariance functions, Weiner processes. |
| Lecture 15 | Orthonormal expansions, Gaussian sinc
processes, filtered Gaussian sinc processes. |
| Lecture 16 | Second-order characterizations of
filtered random processes, Spectral
densities; introduction to estimation. |
| Lecture 17 | MMSE estimation, estimation and Gaussian
random vectors, LLSE estimation, optimal filtering.
|