| 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, conditional distributions,
functions of random variables, expected values.
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| Lecture 4 | Moments, Gaussian random
variables, Markov's inequality, Chebyshev's inequality,
Chernoff bounds, Moment Generating Functions. |
| Lecture 5 | Random vectors, joint cdfs, pdfs and pmfs,
marginal distributions, independence. |
| Lecture 6 | Conditional Distributions, functions
of multiple random variables, sums of random variables,
characteristic functions, linear transformations of
random vectors, expectation and moments of random vectors,
correlation and covariance. |
| Lecture 7 | Correlation coeeficients, jointly Gaussian
random variables, Gaussian Random variables and linear
transformations.
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| Lecture 8 |
Covariance matrices, conditioning and jointly Gaussian Random
variables, conditional expectation, iterated
expectation, introduction to estimation.
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| Lecture 9 |
Baysian MMSE estimation, estimation and
jointly Gaussian random variables, LLSE
estimation. |
| Lecture 10 | Orthogonality property of MMSE estimates,
Discrete-time random processes,
i.i.d. processes and laws of large numbers, mean square convergence,
convergence in probability, Almost sure convergence. |
| MID-TERM EXAM |
| Lecture 11 | Convergence in
distribution, the central limit theorem. |
| Lecture 12 | Finite dimensional distributions and
Kolmogorov's theorem,
Stationary processes, memoryless processes, stationary increments, independent
increments, Markov property, counting processes, random
walks.
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| Lecture 13 | Markov chains: transition
matrices/graphs, n-step transistions, first-step analysis.
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| Lecture 14 | Markov chains: state classification, stationary distribtuions. |
| 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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