**Basic probability:**

Probability space and axioms, basic laws, conditional probability and Bayes rule, independence.

Random variables, probability mass function (pmf), cumulative distribution function (cdf), probability density function (pdf), discrete, continuous, and mixed random variables, functions of random variables, generation of random variables.

Pairs of random variables, joint, marginal, and conditional distributions, maximum likelihood (ML) and maximum a posteriori probability (MAP) detection.

Expectation, mean, variance, characteristic function, covariance and correlation, Markov and Chebychev inequalities, Jensen's inequality, conditional expectation.

Minimum mean square error (MMSE) estimation, linear estimation, jointly Gaussian random variables.

**Random vectors:**

Extension of cdf, pdf, and pmf to more than two random variables, independence and conditional independence, covariance matrix, Gaussian random vectors, linear estimation - vector case.

Modes of convergence, laws of large numbers, central limit theorem.

**Random processes:**

Discrete-time and continuous-time random processes, memoryless, independent increment, Markov, and Gaussian random processes, point processes.

Stationary processes, autocorrelation functions and power spectral density (psd), white noise, bandlimited processes.

Response of linear systems to random inputs, linear estimation - process case, infinite smoothing, Wiener filtering.

This is a tentative schedule for the course. Changes may be made as the
course progresses, and we will update the lecture topics accordingly.

[Apr 01, Mon] Lecture #1: Introduction, aptitude test

[Apr 03, Wed] Lecture #2: Basic probability theory

[Apr 05, Fri] Lecture #3: Basic probability theory contd.

[Apr 09, Mon] Lecture #4: Random variables

[Apr 11, Wed] Lecture #5: Random variables contd.

[Apr 13, Fri] No lecture

[Apr 16, Mon] Lecture #6: Random variables contd.

[Apr 18, Wed] Lecture #7: Pairs of random variables

[Apr 20, Fri] Lecture #8: Random pairs contd.

[Apr 23, Mon] Lecture #9: Random pairs contd.

[Apr 25, Wed] Lecture #10: MAP detection

[Apr 27, Fri] Midterm #1

[Apr 30, Mon] Lecture #11: Expectation

[May 02, Wed] Lecture #12: Expectation contd.

[May 04, Fri] Lecture #13: Conditional expectation

[May 07, Mon] Lecture #14: MMSE estimation

[May 09, Wed] Lecture #15: Linear estimation and jointly Gaussian random variables

[May 11, Fri] Lecture #16: Random vectors

[May 14, Mon] Lecture #17: Random vectors contd.

[May 16, Wed] Lecture #18: Convergence

[May 18, Fri] Midterm #2

[May 21, Mon] Lecture #19: Random processes

[May 23, Wed] Lecture #20: Random processes contd.

[May 25, Fri] Lecture #21: Random processes contd.

[May 28, Mon] No lecture (Memorial Day)

[May 30, Wed] Lecture #22: Markov chains

[Jun 01, Fri] Lecture #23: Stationary processes

[Jun 04, Mon] Lecture #24: Stationary processes contd.

[Jun 06, Wed] Lecture #25: LTI filtering of stationary processes

[Jun 08, Fri] Lecture #26: LTI filtering of stationary processes contd.

[Jun 15, Fri] Final exam