Mathematics III

MA2004E MATHEMATICS III

Course Structure

L	T	P	O	C
3	1*	0	5	3

Total Lecture Sessions: 39

Course Outcomes

• CO1: Introduce the fundamentals of probability and random variables in continuous/discrete settings.
• CO2: Identify the distribution and transformation of random variables.
• CO3: Apply the concepts of correlation and stationarity in analysis of stochastic processes.

Probability distributions of a single random variable

• Basics of probability, Axioms of Probability, Conditional probability, Independence
• Random variables: Discrete and Continuous random variables, Probability Distribution functions, Cumulative Distribution function, Expectation, Variance, Moment Generating Function, Higher Order Moments
• Special Distributions:

   * Binomial distribution
   * Geometric distribution
   * Poisson distribution
   * Hypergeometric distribution
   * Uniform distribution
   * Gamma distribution
   * Exponential distribution
   * Normal distribution

• Markov and Chebyshev inequalities
• Law of large numbers
• Central limit theorem and its significance.

Probability distributions of several random variables

• Joint probability distribution function
• Joint probability mass and density function
• Marginal and Conditional distributions
• Transformation of random variables
• Joint probability distribution of functions of random variables
• Independent random variables
• Covariance
• Correlation coefficient
• Bivariate normal distribution

Random processes

• Introduction and Specification
• Mean and Auto-Correlation Function
• Auto-Covariance Function
• Cross-Correlation and Cross-Covariance Functions
• Stationary processes:

   * Strict-Sense Stationarity
   * Wide-Sense Stationarity (WSS)
   * Stationarity
   * Auto-Correlation Function
   * Cross-Correlation Function
   * Power Spectral Density of a WSS Random process
   * Wiener-Khinchine theorem
   * Low-pass and Band-pass processes
   * Power and Bandwidth calculations

• Time averaging and Ergodicity:

   * Time averages - interpretation, Mean and Variance
   * Ergodicity: general definition, Ergodicity of the mean, Ergodicity of the auto-correlation function.

References

1. Ross, S. M. (2014). *Introduction to Probability and Statistics for Engineers and Scientists* (5th ed.). Academic Press (Elsevier).
2. Johnson, Richard A. (2011). *Miller & Freund’s - Probability and Statistics for Engineers* (8th ed.). Prentice Hall India.
3. Krishnan, V. (2006). *Probability and Random Processes* (2nd ed.). John Wiley & Sons.
4. Ross, S. (2014). *A First Course in Probability* (9th ed.). Pearson.
5. Yates, Roy D., & Goodman, David J. (2021). *Probability and Stochastic Processes* (3rd ed.). Wiley.
6. Miller, Scott, & Childers, Donald. (2007). *Probability and Random Processes* (2nd ed.). Elsevier.