1. Review of probability theory: set theory, probability axioms, conditional probability, independence, random variables, discrete and continuous random variables, cumulative distribution function (CDF), probability mass function (PMF), probability density function (PDF), conditional PMF/PDF, expected value, variance, functions of a random variable, expected value of the derived random variable, multiple random variables (discrete and continuous), joint CDF/PMF/PDF, marginal PMF/PDF, functions of multiple random variables, multiple functions of multiple random variables, independent random variables, uncorrelated random variables.
2. Sums of random variables, moment generating fuction, random sums of random variables.
3. MMSE estimation: blind, linear, unconstraint MMSE estimators, the Gaussian case, orthogonality principle, innovations sequences.
4. The sample mean, laws of large numbers, central limit theorem, convergence of sequence of random variables.
5. Introduction to random processes, specification of random processes, nth order joint PDFs, independent increments, stationary increments, Markov property.
6. Gaussian process, Poisson process and Brownian motion.
7. Mean and correlation of random processes, stationary, wide sense stationary, ergodic processes.
8. Mean-square continuity, mean-square derivatives.
9. Random signal processing: random processes as inputs to linear time invariant systems, power spectral density, Gaussian processes as inputs to LTI systems, white Gaussian noise.
10. MMSE estimation for random processes: whitening filters, prediction, smoothing, Yule-Walker equations, Wiener-Hopf equations, Kalman filtering.
11. Discrete-time Markov chains: state and n-step transition probabilities, Chapman-Kolmogorov equations, first passage probabilities, classification of states, limiting state probabilities.
12. Continuous-time Markov chains.
13. Karhunen-Loeve expansion.