| Topics | Reading |
| Lecture 1 | Introduction,
Probability spaces, Sigma algebras, countable and uncountable sets.
| Sect. 1.1-1.2 |
| Lecture 2 | Properties of Probability measures, Conditional probability, statistical independence, conditional independence, Random variables and CDF.
| Sect. 1.3.1-1.3.3 |
| Lecture 3 | Discrete, continuous, mixed and singular random
variables, Multiple random variables, joint distributions, independence and conditioning for random variables, stochastic processes. | Sect. 1.3.4 -1.4 |
| Lecture 4 | The Bernoulli process, Expected values,
moments, sums of random variables.
| Sect. 1.4- 1.5 |
| Lecture 5 | Conditional expectation, iterated expectation, Markov's
inequality, Chebyshev's inequality, the coupon collector problem. | Sect. 1.5-1.6 |
| Lecture 6 | More on coupon collector problem, Chernoff Bounds, Moment generating functions, Chernoff bound examples.
| Sect. 1.6 |
| Lecture 7 | Moment generating functions and sums of independent random
variables, log-moment generating functions, 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 Strong Law of Large numbers, Convergence
in distribution, the central limit theorem, characteristic functions and CLT proof.
| Sect. 1.7, CLT notes |
| Lecture 9 | Gaussian approximations, Counting processes, The Poisson process: memoryless property, fresh-restart property, increment properties.
| Sect. 2.1-2.2 |
| Lecture 10 | Poisson processes: Distribution of number of arrivals, Baby Bernoulli interpretation, splitting and combining Poisson processes.
| Sect. 2.2-2.3. |
| Lecture 11 | Markov property, Markov Chains: transition
Matrices/Graphs, first-step analysis. | Sect. 4.1, M.C. Notes |
| Lecture 12 | Markov Chains: stationary distributions, State classifications, recurrence, null recurrence | Sect. 4.1-4.2 & 6.2 |
| Lecture 13 | MID-TERM EXAM |
| 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, MMSE estimation. . | Sect. 3.5, Sect. 10.1 - 10.2 |
| Lecture 18 | Properties of MMSE estimators, MMSE with Gaussian random vectors,
linear estimation, Intro. to Gaussian Processes. |
Sect. 10.2-10.3, Sect. 3.6.1 |
| Lecture 19 | Stationary processes, Properties of
covariance functions, Wiener processes. | Sect. 3.6.1,
3.6.9 |