UNIT I THE RANDOM VARIABLE

1.1 THE RANDOM VARIABLE : Introduction, Definition of a Random Variable, Conditions for a Function to be a Random Variable, Discrete, Continuous and Mixed Random Variables

1.2 Distribution and Density functions, Properties

1.3 Binomial, Poisson, Uniform, Gaussian, Exponential,Rayleigh, Conditional Distribution, Conditional Density, Properties

UNIT II OPERATION ON ONE RANDOM VARIABLE – EXPECTATIONS

2.1 OPERATION ON ONE RANDOM VARIABLE – EXPECTATIONS : Introduction, Expected Value of a Random Variable, Function of a Random Variable

2.2 Moments about the Origin, Central Moments, Variance and Skew

2.3 Chebyshev’s Inequality, Characteristic Function, Moment Generating Function

2.4 Transformations of a Random Variable, Monotonic Transformations for a Continuous Random Variable, Non-monotonic Transformations of Continuous Random Variable

UNIT III MULTIPLE RANDOM VARIABLES

3.1 MULTIPLE RANDOM VARIABLES : Vector Random Variables, Joint Distribution Function, Properties of Joint Distribution

3.2 Marginal Distribution Functions, Conditional Distribution and Density

3.3 Statistical Independence, Sum of Two Random Variables, Sum of Several Random Variables, Central Limit Theorem: Unequal Distribution, Equal Distributions

3.4 OPERATIONS ON MULTIPLE RANDOM VARIABLES: Joint Moments about the Origin, Joint Central Moments, Joint Characteristic Functions

3.5 Jointly Gaussian Random Variables: Two Random Variables case, N Random Variables case, Properties, Transformations of Multiple Random Variables, Linear Transformations of Gaussian Random Variables

UNIT IV RANDOM PROCESSES – TEMPORAL CHARACTERISTICS

4.1 RANDOM PROCESSES – TEMPORAL CHARACTERISTICS: The Random Process Concept, Classification of Processes, Deterministic and Nondeterministic Processes Distribution and Density Functions

4.2 Concept of Stationarity and Statistical Independence, First-Order Stationary Processes, Second- Order and Wide-Sense Stationarity

4.3 Nth-order and Strict-Sense Stationarity, Time Averages and Ergodicity, Autocorrelation Function and its Properties, Cross-Correlation Function and its Properties, Covariance Function

4.4 Gaussian Random Processes, Poisson Random Process

UNIT V RANDOM PROCESSES – SPECTRAL CHARACTERISTICS

5.1 RANDOM PROCESSES – SPECTRAL CHARACTERISTICS: The Power Spectrum: Properties, Relationship between Power Spectrum and Autocorrelation Function

5.2 The Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Spectrum and Cross-Correlation Function.

UNIT VI LINEAR SYSTEMS WITH RANDOM INPUTS

6.1 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

6.2 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

6.3 Modeling of Noise Sources: Resistive (Thermal) Noise Source, Arbitrary Noise Sources, Effective Noise Temperature, Average Noise Figure, Average Noise Figure of cascaded networks