| 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, PMFs, PDFs, mixed and singular random
variables.
| Sect. 1.3.1-1.3.3 |
| Lecture 3 | Multiple random variables, joint
distributions, stochastic processes, the Bernoulli process. | Sect. 1.3.4 -1.4 |
| Lecture 4 | Expected values,
moments, sums of random variables, conditional expectation,
The coupon collector problem. | Sect. 1.5 |
| Lecture 5 | Markov's
inequality, Chebyshev's inequality, Chernov Bounds. | Sect. 1.6 |
| Lecture 6 |
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. | Sect. 1.6-1.7 |
Lecture 7 | Almost sure convergence, the strong law of large
numbers, convergence
in distribution,, the central limit theorem. | Sect. 1.7 & 5.2 |
| Lecture 8 | Central limit theorem examples,
Characteristic functions, Counting Processes and the Poisson
Process, memoryless property, fresh-restart property. | Sect. 2.1-2.2 |
| Lecture 9 | More on Poisson Processes: increment
properties, alternative definitions,
Splitting and combining Poisson Processes. | Sect. 2.2-2.3 |
| Lecture 10 | Markov Chains: definition, transition
Matrices/Graphs, first-step analysis. | Notes |
| Lecture 11 | Markov Chains: Stationary
distributions, state classifications, recurrence. | Sect. 4.1-4.2 & 6.2 |
| Lecture 12 | MID-TERM EXAM |
| Lecture 13 | Markov Chains: null recurrence, periodicity, ergodic chains, convergence to stationary
distributions, balance equations. | Sect. 4.2-4.3, 6.1-6.2 |
| Lecture 14 | Gaussian random vectors: linear
transformations, moment generating functions. | Sect. 3.1 -3.3 |
| Lecture 15 | Gaussian random vectors: joint probability
distributions and properties of covariance matrices | Sect. 3.3 -3.4 |
| | Lecture 16 | Conditioning and Gaussian random vectors;
Introduction to Estimation | Sect. 3.5, Sect. 10.1. |
| Lecture 17 | Estimation and Gaussian random vectors,
linear estimation. Intro. to Gaussian Processes. |
Sect. 10.2-10.3. |
| Lecture 18 | Stationary processes, Properties of
covariance functions, Weiner processes. | Sect. 3.6.1, 3.6.9 |