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## Random Variables and Stochastic Process

### OBJECTIVES:

• to grant students AN introduction to elementary applied math, in preparation for courses on applied math analysis, random variables and random processes.

• To mathematically model the random phenomena with the assistance of applied math ideas.

• To introduce the necessary ideas of random variables and random processes.

• to research the LTI systems with stationary random method as input.

• To introduce the categories of noise and modelling noise sources.

#### UNIT I

THE variate : Introduction, Review of applied math, Definition of a variate, Conditions for a operate to be a variate, Discrete, Continuous and Mixed Random Variables, Distribution and Density functions, Properties, Binomial, Poisson, Uniform, Gaussian, Exponential, Rayleigh, Conditional Distribution, Conditional Density, Properties.

#### UNIT II

OPERATION ON ONE variate – EXPECTATIONS : Introduction, first moment of a variateoperate of a variate, Moments concerning the Origin, Central Moments, Variance and Skew, Chebychev’s difference, Characteristic operate, Moment Generating operate, Transformations of a variate: Monotonic Transformations for endless Random Variable, nonmonotonic Transformations of Continuous variate

#### UNIT III

MULTIPLE RANDOM VARIABLES : Vector Random Variables, Joint Distribution operate, Properties of Joint Distribution, Marginal Distribution Functions, Conditional Distribution and Density, applied math Independence, add of 2 Random Variables, add of many Random Variables, Central Limit Theorem: Unequal Distribution, Equal Distributions. OPERATIONS ON MULTIPLE RANDOM VARIABLES: Joint Moments concerning the Origin, Joint Central Moments, Joint Characteristic Functions, collectivelymathematician Random Variables: 2 Random Variables case, N Random Variables case, Properties, Transformations of Multiple Random Variables, Linear Transformations of mathematician Random Variables.

#### UNIT IV

RANDOM PROCESSES – TEMPORAL CHARACTERISTICS: The Random method idea, Classification of Processes, settled and Nondeterministic Processes, Distribution and Density Functions, idea of Stationarity and applied math Independence. First-Order Stationary Processes, Second-order and Wide-Sense Stationarity, Nth-order and Strict-Sense Stationarity, Time Averages and haphazardness, Autocorrelation operate and its Properties, Cross-Correlation operate and its Properties, variance Functions, mathematician Random Processes, Poisson Random method

#### UNIT V

RANDOM PROCESSES – SPECTRAL CHARACTERISTICS: the facility Density Spectrum: Properties, Relationship between Power Density Spectrum and Autocorrelation operate, The Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Density Spectrum and Cross-Correlation operate

II Year – I Semester
L T P C
4 0 0 3
RANDOM VARIABLES PROCESSES

#### UNIT VI

LINEAR SYSTEMS WITH RANDOM INPUTS : Random Signal Response of Linear Systems: System Response – Convolution, Mean and Mean-squared worth of System Response, Autocorrelation operate of Response, Cross-Correlation Functions of Input and Output, Spectral Characteristics of System Response: Power Density Spectrum of Response, Cross-Power Density Spectra of Input and Output, Band pass, Band-Limited and Narrowband Processes, Properties, Modeling of Noise supplys: Resistive (Thermal) Noise Source, impulsive Noise Sources, Effective Noise Temperature, Average Noise Figure, Average Noise Figure of cascaded networks.

#### TEXT BOOKS:

onelikelihood, Random Variables & Random Signal Principles, Peyton Z. Peebles, TMH, fourthEdition, 2001. 2. likelihood, Random Variables and random Processes, Athanasios Papoulis and S.Unnikrisha, PHI, fourth Edition, 2002.

#### REFERENCE BOOKS:

oneapplied math and random Processes – B. Prabhakara Rao, baccalaureate Publications

twolikelihood and Random Processes with Applications to Signal process, Henry Stark and John W. Woods, Pearson Education, third Edition.

3. Schaum’s define of likelihood, Random Variables, and Random Processes.

4. AN Introduction to Random Signals and discipline, B.P. Lathi, International Textbook, 1968.

5. Random method – Ludeman , John Wiley half-dozenapplied math and Random Processes, P. Ramesh adult male, McGrawHill, 2015.

#### OUTCOMES:

After completion of the course, the scholarareready to

• Mathematically model the random phenomena and solve easy probabilistic issues.

• establishdiffering types of random variables and reckonapplied math averages of those random variables.

• Characterize the random processes within the time and frequency domains.

• Analyze the LTI systems with random inputs.

• Apply these techniques to research the systems within the presence of various kinds of noise.