| Topics | Reading |
| Lecture 1 | Introduction,
Probability spaces, Sigma Algebras, Properties of
probability measures,
| Sect. 1.1-1.2 |
| Lecture 2 | Conditional probability, statistical independence,
conditional independence, repeated trials,
random variables, CDFs.
| Sect. 1.3.1-1.3.3 |
| Lecture 3 | PMFs, PDFs, mixed and singular random
variables, multiple random variables, joint
distributions, independence and conditioning. | Sect. 1.3.4 -1.4 |
| Lecture 4 | Stochastic processes, the Bernoulli process, expected values,
moments, sums of random variables.
| Sect. 1.5 |
| Lecture 5 | Conditional expectation, Markov's
inequality, Chebyshev's inequality. | Sect. 1.6 |
| Lecture 6 |
The coupon collector problem, Chernov Bounds.
Moment generating functions and sums of independent random variables, log-moment
generating functions. | Sect. 1.6-1.7 |
| Lecture 7 | Convergence of random variables: mean-squared
convergence, convergence in probability, the weak law of large numbers, Almost sure convergence, the strong law of large
numbers. | Sect. 1.7 & 5.2 |
| Lecture 8 | Proof of the strong law of large numbers, convergence
in distribution, the central limit theorem, characteristic functions. | Sect. 1.7 & 5.2 |
| Lecture 9 | Counting processes and the Poisson process, memoryless property, fresh-restart property, increment properties,
| Sect. 2.1-2.2 |
| Lecture 10 | Poisson processes: Distribution of number of arrivals, alternative definitions, splitting and combining Poisson processes. | Sect. 2.2-2.3 |
| Lecture 11 | MID-TERM EXAM |
| Lecture 12 | Markov Chains: definitions, transition
Matrices/Graphs, first-step analysis | Notes |
| Lecture 13 | Markov Chains: Stationary
distributions, state classifications, recurrence, null recurrence | Sect. 4.1-4.2 & 6.2 |
| Lecture 14 | Markov Chains: periodicity, ergodic chains, convergence to stationary distributions, balance equations. | Sect. 4.2-4.3, 6.1-6.2 |
| Lecture 15 | Gaussian random vectors: linear
transformations, moment generating functions. | Sect. 3.1 -3.3 |
| Lecture 16 | Gaussian random vectors: joint probability
distributions and properties of covariance matrices | Sect. 3.3 -3.4 |
| | Lecture 17 | Conditioning and Gaussian random vectors;
Introduction to estimation | Sect. 3.5, Sect. 10.1. |
| Lecture 18 | Estimation and Gaussian random vectors,
linear estimation, Intro. to Gaussian Processes. |
Sect. 10.2-10.3. |
| Lecture 19 | Stationary processes, Properties of
covariance functions, Wiener processes. | Sect. 3.6.1,
3.6.9 |