Probability spaces, properties of probability measures,
conditional probability, independence.
|Lecture 2||Bayes' rule and
inference, independent trials, discrete random variables
and probability mass functions, cumulative distribution
functions, continuous random variables and densities.
| Lecture 3 ||Mixed and singular
random variables, conditional distributions,
functions of random variables, expected values.
|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
||Covariance matrices, conditioning and jointly Gaussian Random
variables, conditional expectation, iterated
expectation, introduction to estimation.
|| Baysian MMSE estimation, estimation and
jointly Gaussian random variables, LLSE
|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
Stationary processes, memoryless processes, stationary increments, independent
increments, Markov property, counting processes, random
|Lecture 13||Markov chains: transition
matrices/graphs, n-step transistions, first-step analysis.
|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
|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.