Module-1 Single Random Variables : Definition of random variables, cumulative distribution function continuous and discrete random variables; probability mass function, probability density functions and properties; Expectations, Characteristic functions, Functions of single Random Variables, Conditioned Random variables. Application exercises to some special distributions : Uniform, Exponential, Laplace, Gaussian; Binomial, and Poisson distribution. (Chapter - 1) Module-2 Multiple Random Variables : Concept, Two variable CDF and PDF, Two Variable expectations (Correlation, Orthogonality, Independent), Two variable transformation, Two Gaussian Random variables, Sum of two independent Random Variables, Sum of IID Random Variables - Central limit Theorem and law of large numbers, Conditional joint Probabilities, Application exercises to Chi-square RV, Student-T RV, Cauchy and Rayleigh RVs. (Chapter - 2) Module-3 Random Processes : Ensemble, PDF, Independence, Expectations, Stationarity, Correlation Functions (ACF, CCF, Addition, and Multiplication), Ergodic Random Processes, Power Spectral Densities (Wiener Khinchin, Addition and Multiplication of RPs, Cross spectral densities), Linear Systems (Output Mean, Cross correlation and Auto correlation of Input and output), Exercises with Noise. (Chapter - 3) Module-4 Vector Spaces : Vector spaces and Null subspaces, Rank and Row reduced form, Independence, Basis and dimension, Dimensions of the four subspaces, Rank-Nullity Theorem, Linear Transformations Orthogonality : Orthogonal Vectors and Subspaces, Projections and Least squares, Orthogonal Bases and Gram- Schmidt Orthogonalization procedure. (Chapters - 4, 5) Module-5 Determinants : Properties of Determinants, Permutations and Cofactors. Eigen values and Eigen vectors : Review of Eigen values and Diagonalization of a Matrix, Special Matrices (Positive Definite, Symmetric) and their properties, Singular Value Decomposition. (Chapters - 6, 7)
