### PROBABILITY THEORY AND STOCHASTIC PROCESSES syllabus

PROBABILITY THEORY AND STOCHASTIC PROCESSES
UNIT I
PROBABILITY : 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, Total Probability, Bayes Theorem, Independent Events:
UNIT II
THE RANDOM VARIABLE : Definition of a Random Variable, Conditions for a Function to be a Random Variable, Discrete and Continuous, Mixed Random Variable, Distribution and Density functions, Properties, Binomial, Poisson, Uniform, Gaussian, Exponential, Rayleigh, Conditional Distribution, Methods of defining Conditioning Event, Conditional Density, Properties.
UNIT III
OPERATION ON ONE RANDOM VARIABLE EXPECTATIONS : Introduction, Expected Value of a Random Variable, Function of a Random Variable, Moments about the Origin, Central Moments, Variance and Skew, Chebychevs Inequality, Characteristic Function, Moment Generating Function, Transformations of a Random Variable: Monotonic Transformations for a Continuous Random Variable, Nonmonotonic Transformations of Continuous Random Variable, Transformation of a Discrete Random Variable.
UNIT IV
MULTIPLE RANDOM VARIABLES : Vector Random Variables, Joint Distribution Function, Properties of Joint Distribution, 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, (Proof not expected). Unequal Distribution, Equal Distributions.
UNIT V
OPERATIONS ON MULTIPLE RANDOM VARIABLES : 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 VI
STOCHASTIC PROCESSES TEMPORAL CHARACTERISTICS : The Random Process Concept, Classification of Processes, Deterministic and Nondeterministic Processes, Distribution and Density Functions, concept of Stationarity and Statistical Independence. First-Order Stationary Processes, Second- Order and Wide-Sense Stationarity, (N-Order) and Strict-Sense Stationarity, Time Averages and Ergodicity, Mean-Ergodic Processes, Correlation-Ergodic Processes, Autocorrelation Function and Its Properties, Cross-Correlation Function and Its Properties, Covariance Functions, Gaussian Random Processes, Poisson Random Process.
UNIT VII
STOCHASTIC PROCESSES SPECTRAL CHARACTERISTICS: The Power Spectrum: Properties, Relationship between Power Spectrum and Autocorrelation Function, the Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Spectrum and Cross-Correlation Function.
UNIT VIII: NOICE:
Types of noise: resistive ( thermal ) noise source. Shot noise, extra terrestrial noise. Arbitrary Noise Sources, white noise, narrow band noise, In phase and quadrature phase components and its properties. Modeling of noise sources, average noise bandwidth. Effective noise temperature. Average noise figures of cascaded networks.
TEXT BOOKS :
1. Probability, Random Variables & Random Signal Principles - Peyton Z. Peebles, TMH, 4th Edition, 2001.
2. Probability, Random Variables and Stochastic Processes Athanasios Papoulis and S. Unnikrishna Pillai, PHI, 4th Edition, 2002.
3. principles of communication systems H Taub, Donald L Schilling, Goutham Saha 2007 TMH
REFERENCES :
1. theory of probability and stochastic processes by pradeep kumar ghosh
2. probability theory and stochastic processes by mallikarjun reddy.
3. Probability and Random Processes with application to signal processing Henry stark and john w woods 3ed PE
4. Probability Methods of signal and system analysis George r cooper clave d mc giflem 3ed 1999 oxford.
5. Statistical theory of communication sp Eugene Xavier 1997 new age publications.