Random Variables and Stochastic Processes

 UNIT-1:INTRODUCTION

Modules for learning

1. Review of Probability Theory, Definition of a Random Variable, 
2. Conditions for a Function to be a Random Variable.
3. Discrete, Continuous and Mixed Random Variable
4.  Distribution and Density functions
5. Properties
6. Binomial
7. Poisson.
8. Uniform
9. Gaussian, Exponential
10. Rayleigh
11. Conditional Distribution
12. Conditional Density
13. Properties

UNIT-2: OPERATION ON ONE RANDOM VARIABLE - EXPECTATIONS

Modules for learning

1. Introduction.
2. Expected Value of a Random Variable, Function of a Random Variable.
3. Moments about the Origin.
4. Central Moments, Variance and Skew
5. Chebyshev’s Inequality, Characteristic Function.
6. Transformations of a Random Variable.
7. Monotonic Transformations for a Continuous Random Variable.
8. Non-monotonic Transformations of Continuous Random Variable.

UNIT-3: MULTIPLE RANDOM VARIABLES

Modules for learning

1. Vector Random Variables.
2. Joint  Distribution Function.
3. Properties of Joint Distribution, Marginal Distribution Functions.
4. Conditional Distribution and Density.
5. Statistical Independence, Sum of Two Random Variables.
6. Sum of Several Random Variables.
7. Central Limit Theorem: Unequal Distribution.
8. Central Limit Theorem:  Equal Distributions
9. OPERATIONS ON MULTIPLE RANDOM VARIABLES: Joint Moments about the Origin.
10. Joint Central Moments, Joint Characteristic Functions.
11. Jointly Gaussian Random Variables: Two Random Variables case.
12. Transformations of Multiple Random Variables.
13. Linear Transformations of Gaussian Random Variables.

UNIT-4: RANDOM   PROCESSES   –  TEMPORAL  CHARACTERISTICS

Modules for learning

1. The Random Process Concept. Slides
2. Distribution and Density Functions. Slides
3. Statistically Independent random process and Independence of two random processes. Slides
4. Concept of Stationarity, First-Order, Second-order, N-th order Stationary Processes, strict sense Stationarity. Slides
5. Wide-Sense Stationarity. Slides
6. Time Averages and Ergodicity. Slides
7. Autocorrelation Function and its Properties. Slides
8. Cross-Correlation Function and its Properties. Slides
9. Covariance Functions. Slides
10. Gaussian Random Processes. Slides
11. Poisson Random Process. Slides
12. Classification of Processes - Deterministic and Nondeterministic Processes. Slides

UNIT-5: RANDOM PROCESSES   - SPECTRAL CHARACTERISTICS

Modules for learning

1. The Power Density Spectrum. (Part A)
2. The Power Density Spectrum (Part B)
3. Properties.
4. Relationship between Power Density Spectrum and Autocorrelation Function.
5. The Cross-Power Density Spectrum, Properties.
6. Revisit Correlation of Processes - WSS and Ergodic, Graphical illustration of Cross-Correlation
7. Relationship between Cross-Power Density Spectrum and Cross-Correlation Function

UNIT-6: LINEAR SYSTEMS WITH RANDOM INPUTS

Modules for learning

1. Random Signal Response of Linear Systems: System Response
2. System Response – Convolution, Mean and Mean-squared Value of System Response
3. Autocorrelation Function of Response.
4. Cross-Correlation Functions of Input and Output
5. Cross-Power Density Spectrum: Exercises
6. Spectral characteristics of systems - Response: Power Density Spectrum of Response
7. Cross-Power Density Spectra of Input and Output
8. Band pass,  Band-Limited and Narrow-band Processes, Properties
9. Properties of Band-Limited Processes