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GR15 Regulations (2015-16) GOKARAJU RANGARAJU INSTITUTE OF ENGINEERING AND TECHNOLOGY PROBABILITY THEORY AND STOCHASTIC PROCESSES Course Code: GR15A2050 II Year I Semester L:3 T:1 P:0 C:4 Prerequisites • Fundamentals of Probability theory • Basics of Fourier series, transforms Course Objectives • To understand the fundamentals of probability • To understand the principles of random signals and random processes • To be acquainted with systems involving random signals, Noise concepts and data coding • To know random phenomena which occur in engineering applications Course Outcomes • Interpret probability by modeling sample spaces. • Construct the probability distribution of a random variable, based on a real-world situation, and use it to compute expectation and variance • Evaluate response of a linear system to Random Process Unit-I PROBABILITY & RANDOM VARIABLES Probability introduced through Sets and Relative Frequency: Experiments and Sample Spaces, Discrete and Continuous Sample Spaces, Events, Probability Definitions and Axioms, Mathematical Model of Experiments, Probability as a Relative Frequency, Joint Probability, Conditional Probability, Baye’s Theorem, Independent Events, Random Variable, Functions of random variable, Discrete and Continuous, Mixed Random Variable, Distribution and Density functions, Binomial, Poisson, Uniform, Gaussian Distribution. Unit-II OPERATIONS ON SINGLE VARIABLE – EXPECTATIONS Introduction, 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, Transformation of a Discrete Random Variable. Vector Random Variables 65 GR15 Regulations (2015-16) Unit-III OPERATIONS ON & MULTIPLE RANDOM– EXPECTATIONS Joint Distribution Function, Properties of Joint Distribution, Marginal Distribution Functions, Conditional Distribution and Density - Point Conditioning, Conditional Distribution and Density -Statistical Independence, Sum of Two Random Variables, Sum of Several Random Variables, Central Limit Theorem, Expected Value of a Function of Random Variables: Joint Moments about the Origin, Joint Central Moments, Joint Characteristic Functions. Unit-IV RANDOM PROCESSES -TEMPORAL CHARACTERISTICS The Random process, classification, deterministic and no ndeterministic processes, distribution and density Functions, stationarity and statistical independence, first-order stationary processes, second-order and wide-sense stationarity, auto correlation function and its properties, cross-correlation function and its properties, covariance functions, Gaussian random processes, random signal response of linear systems, autocorrelation and cross-correlation functions of input and output. Unit-V RANDOM PROCESSES-SPECTRAL CHARACTERISTICS AND NOISE The Power Spectrum: Properties, Relationship between Power Spectrum and Autocorrelation Function, Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Spectrum and Cross-Correlation Function. Spectral Characteristics of System Response: Power Density Spectrum of Response, Cross-Power Density Spectrums of Input and Output. MODELLING OF NOISE Introduction to noise, types and sources of noises, noise in communication system, Arbitrary Noise Sources, Resistive and Thermal Noise Source, Effective Noise Temperature, Average Noise Figures, Average Noise Figure of cascaded networks. Teaching methodologies • Power Point presentations • Tutorial Sheets • Assignments • Experiments with Matlab software Text Books 1. Probability, Random Variables and Stochastic Processes - Athanasios Papoulis and S. Unnikrishna Pillai, PHI, 4th Edition, 2002. 2. Probability, Random Variables & Random Signal Principles - Peyton Z. Peebles, TMH, 4th Edition, 2001 66 GR15 Regulations (2015-16) Reference Books 1. Probabilistic Methods of Signals and System Analysis-George R. Cooper and Clare D. McGillem, oxford, 3rd Edition, 2007. 2. R.P. Singh and S.D. Sapre, “Communication Systems Analog & Digital”, 1995,TMH. Web Resources 1. Http://walrandpc.eecs.berkeley.edu/126notes.pdf 2. http://nptel.iitm.ac.in/courses/Webcoursecontents/IIT%20Guwahati/ Probability_rp/index.htm 67