**Download/Source url :**

http://www.youtube.com/watch?v=p6lze5nPkzk&feature=player_embedded

**By Prof.Bikash. Kumar. Dey , Department of Electrical Engineering,IIT Bombay.**

**PROBABILITY THEORY AND STOCHASTIC PROCESSES SYLLABUS :**

**Source :http://jntuk.edu.in**

**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, Chebychev’s 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**

__RANDOM 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**

__RANDOM 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**

__LINEAR SYSTEMS WITH RANDOM INPUTS :__Random Signal Response of Linear Systems: System Response – Convolution, Mean and Mean-squared Value of System Response, autocorrelation Function of Response, Cross-Correlation Functions of Input and Output, Spectral Characteristics of System Response:

Power Density Spectrum of Response, Cross-Power Density Spectrums of Input and Output, Band pass,

Band-Limited and Narrowband Processes, Properties, Modeling of Noise Sources: Resistive (Thermal) Noise

Source, Arbitrary Noise Sources, Effective Noise Temperature, Average Noise Figures, Average Noise Figure

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.

**REFERENCES :**

1. Communication Systems Analog & Digital – R.P. Singh and S.D. Sapre, TMH, 1995.

2. Probability and Random Processes with Application to Signal Processing – Henry Stark and John W.

Woods, Pearson Education, 3rd Edition.

3. Probability Methods of Signal and System Analysis. George R. Cooper, Clave D. MC Gillem,

Oxford, 3rd Edition, 1999.

4. Statistical Theory of Communication - S.P. Eugene Xavier, New Age Publications, 2003.

5. Signals, Systems & Communications - B.P. Lathi, B.S. Publications, 2003.

**TTER**
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