| Lecture 1 | Introduction,
Probability spaces.
|
| Lecture 2 | Properties of probability measures,
conditional probabilities, independence. |
| Lecture 3 | Independent trials, combinations,
permutations. |
| Lecture 4 | Random variables, distributions, discrete
Random variables, p.m.f.'s |
| Lecture 5 | Continuous random variables, densities,
mixed random variables, examples of common
distributions. |
| Lecture 6 | Gaussian random variables, conditional distributions, functions of
random variables. |
| Lecture 7 | Expected values, moments. |
| Lecture 8 | Markov inequality, Chebyshev's inequality,
Chernoff bounds, moment generating functions,
Characteristic functions. |
| Lecture 9 | Introduction to random vectors, joint
distributions, joint densities, marginals. |
| Lecture 10 | More on Random vectors: independence,
moments, correlation. |
| Lecture 11 | Two jointly Gaussian random variables,
functions of random vectors. |
| Lecture 12 | Linear transformations of random vectors,
condition distributions. |
| Lecture 13 | More on conditional distributions,
conditional expectation, MMSE estimation. |
| Lecture 14 | More on MMSE estimation; scalar Gaussian
case. |
| Lecture 15 | MMSE estimation - vector case; jointly
Gaussian Random variables (N>2). |
| Lecture 16 | More on Jointly Gaussian random variables
and MMSE estimation;
covariance matrices. |
| Lecture 17 | Laws of large numbers, mean-square
convergence, convergence in probability. |
| Lecture 18 | Almost sure convergence; strong law of
large numbers; convergence in distribution; the central limit theorem. |
| Lecture 19 | More on the central limit theorem; random processes. |
| Lecture 20 | Discrete-time Random processes: Bernoulli
processes, Binomial counting process, simple random walk;
stationarity, Memoryless and Markov properties,
independent and stationary increments. |
| Lecture 21 | Random walk on a graph; Poisson
processes. |
| Lecture 22 | More on Poisson processes; random telegraph
signals. |
| Lecture 23 | Formally specifying random processes,
Kolmorogorov's consistency conditions; mean and
correlation/covariance functions; wide sense
stationarity. |
| Lecture 24 | Properties of covariance functions;
Brownian motion. |
| Lecture 25 | Mean-squared continuity, mean-squared
derivatives. |
| Lecture 26 | Mean-squared integration; random processes
and linear systems. |
| Lecture 27 | Spectral analysis for random
processes; application to linear systems. |
| Lecture 28 | Introduction to optimal filtering; overview
of related courses. |