| Lecture 1 | Introduction,
Probability spaces, properties of probability measures,
conditional probability, independence.
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| Lecture 2 | Bayes' rule and
inference, independent trials, discrete random variables
and probability mass functions, cumulative distribution
functions, continuous random variables and densities.
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| Lecture 3 | Mixed and Singular
Random Variables, properties of density functions,
Conditional Distributions, functions of random variables,
expected values and moments. |
| Lecture 4 | Gaussian Random
Variables, Markov's Inequality, Chebyshev's Inequality,
Chernoff Bounds, Moment Generating Functions.y |
| Lecture 5 | Characteristic Functions; random vectors,
joint cdfs, pdfs and pmfs, marginal distributions. |
| Lecture 6 | Independence of random variables, Functions
of multiple random variables, linear transformations of
random vectors, Expectation and moments of random vectors,
correlation and covariance. |
| Lecture 7 | Linear estimation; Jointly Gaussian pairs of
random variables and
their properties. |
| Lecture 8 | Conditional probability distributions,
Conditional expectation, iterated expectations. |
| Lecture 9 | Baysian MMSE estimation, estimation and
jointly Gaussian random variables. |
| Lecture 10 | Jointly Gaussian random vectors,
Covariance matrices, MMSE estimation for random
vectors. |
| Lecture 11 | Mean-square convergence, convergence in
probability, convergence almost surely, Weak and Strong
Laws of Large numbers |
| Lecture 12 | Convergence in distribution, the central
limit theorem; introduction to stochastic processes. |
| Lecture 13 | Finite dimensional distributions and
Kolmogorov's theorem,
Stationary processes, memoryless processes, stationary increments, independent
increments, Markov property, counting processes, random
walks. |
| Lecture 14 | Markov chains, transition
matrices/graphs, n-step transistions, first-step analysis, stationary
distributions. |
| Lecture 15 | Arrival processes/counting processes,
Poisson processes. |
| Lecture 16 | Mean and correlation/covariance functions,
wide sense stationary processes, Gaussian Processes, Wiener
processes. |
| Lecture 17 | Multiple random processes, cross
correlation functions, Mean-square calculus. |
| Lecture 18 | Mean-square integration, random processes
and linear systems, power spectral density functions. |
| Lecture 19 | Systems driven by white noise; optimal linear
filtering; the non-causal Wiener filter; overview of related courses.
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