Elements of Probability and Statistics

Syllabus Details

• Unit I:(Probability and Random Variables) Random experiments, sample spaces, sets, events, σ-algebras, probability; elements of combinatorial analysis, classical definition and calculation of probability, independence of events, conditional probability, Bayes theorem; random variables, distribution functions, standard discrete and continuous distributions.
• Unit II:(Expectation, Inequalities and Functions of Random Variables) Expectations, raw and central moments, probability and moment generating functions; compound, truncated and mixture distributions; inequalities: Markov, Hölder, Minkowski, Jensen and Liapunov inequalities, etc.; failure rate and its use in defining classes of distributions (e.g. aging classes), series and parallel systems; functions of random variables and their distributions using Jacobian of transformation and other tools.
• Unit III:(Multivariate Distributions and Sampling Distributions) Bivariate CDF (non-standard examples), bivariate moment generating function; bivariate normal, multivariate normal; joint, marginal and conditional PMFs and PDFs; conditional expectation, correlation, multiple and partial correlation; linear and multiple regression; sampling distributions: non-central chi-square, t- and F-distributions and their properties; distributions of quadratic forms under normality and related distribution theory; Fisher–Cochran theorem; examples of use of sampling distributions.
• Unit IV:(Order Statistics and Limit Theorems) Order statistics, CI for p-th percentile (using order statistics), order statistics from standard distributions; modes of convergence, laws of large numbers, CLT.

Course Learning Objective

1. Demonstrate understanding of uncertainty and identify it in nature and in other phenomena and processes.
2. Display the sample space for any random experiment.
3. Explain what a random variable is, recognize different random variables in nature and phenomena, and identify parameters.
4. Determine probabilities of events and probability distributions of random variables.
5. Determine whether or not moments exist for any given random variable and, if so, determine them.
6. Apply tools like probability generating functions and moment generating functions, when they exist, to study distributions of functions of random variables.
7. Discuss and derive distributions of functions of random samples (statistics) from different populations.
8. Explain what statistics and their distributions (sampling distributions) are, and derive distributions of the sample mean, sample variance and order statistics from some standard population distributions.
9. Demonstrate an understanding of different modes of convergence of sequences of random variables, and understand what the laws of large numbers and the Central Limit Theorem say and how they are used.

Recommended Books

1. Dudewicz, E. J. and Mishra, S. N. Modern Mathematical Statistics. Wiley, International Students’ Edition, 1988
2. Rohatgi, V. K. An Introduction to Probability Theory and Mathematical Statistics. Wiley Eastern, 1984
3. Rao, C. R. Linear Statistical Inference and Its Applications. 2nd ed. Wiley Eastern, 1973
4. Pitman, J. Probability. Narosa Publishing House, 1993
5. Johnson, N. L. and Kotz, S. Distributions in Statistics. Vols. I, II and III. Houghton Mifflin, 1972
6. Cramér, H. Mathematical Methods of Statistics. Princeton, 1946