Instructor: Dr. Ruchi Tripathi
Prerequisites: Basic Maths
Class lecture slides along with additional material will be provided over here.
Reading Material [M1] [M2]
Advanced Exercises : B. Hajek Lecture Notes on Probability B. Hajek Lecture Notes on Random Processes , R.~Gallager Lecture Notes on Random Processes
Syllabus
UNIT-I Introduction to Probability: Set theory, Experiments and sample spaces, Discrete and continuous sample spaces, Events, Probability definitions and axioms, Mathematical model of experiments, Joint probability, Conditional probability, Total probability, Bayes’ theorem and Independent events, Bernoulli’s trials.
UNIT-II Random Variables : Definition, Conditions for a function to be a random variable, Discrete, continuous and mixed random variable, Distribution and density functions, Properties, Binomial, Poisson, Uniform, Gaussian, Exponential, Rayleigh, Methods of defining conditioning event, Conditional distribution, Conditional density and their properties, Operation on one random variable: Expected value of a random variable, Function of a random variable, Moments about the origin, Central moments, Variance and skew, Characteristic function,Moment generating function, Transformations of a random variable, Monotonic transformations for a continuous random variable, Non monotonic transformations of continuous random variable, Transformations of discrete random variable.
UNIT-III Multiple Random Variables: Vector random variables, Joint distribution function and its properties, Marginal distribution functions, Conditional distribution and density-point conditioning, Conditional distribution and density-Interval conditioning, Statistical independence, Sum of two random variables, Sum of several random variables, Central limit theorem, Unequal distribution, Equal distributions, Expected value of a function of random variables: Joint moments about the origin, Joint central moments, Joint characteristic functions, Jointly Gaussian random variables: Two random variables case, N random variable case, properties, Transformations of multiple random variables, Linear transformations of Gaussian random variables.
UNIT-IV Stochastic Processes-Temporal Characteristics: The The stochastic process concept, Classification of processes, Deterministic and nondeterministic processes, Distribution and density functions, Statistical independence and concept of stationary: First-order stationary processes, Second order and wide-sense stationarity, Nth order and strict-sense stationary, Time averages and ergodicity, Mean-ergodic processes, Correlation-ergodic processes, Autocorrelation function and its properties, Cross-correlation function and its properties, Covariance functions and its properties, Gaussian random processes, Linear system response: Mean and mean-squared value, Autocorrelation, Cross-correlation functions.
UNIT-V Stochastic Processes-Spectral Characteristics: The power spectrum and its properties, Relationship between power spectrum and autocorrelation function, The cross-power density spectrum and properties, Relationship between cross-power spectrum and cross-correlation function. Spectral characteristics of system response: Power density spectrum of response, Cross power spectral density of input and output of a linear system.
Reference books:
A. Papoulis and S. U. Pillai, Probability Random Variables and Stochastic Processes, 4th edition. McGraw-Hill, 2002.
H. Stark and J. W. Woods, Probability and Random Processes with Applications to Signal Processing,4th edition, Pearson,2011.
A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 3rd edition, Pearson, 2011.