Probability Theory & Statistics

The objective of this course is to enable students a) to learn formal treatment of

probability theory and its applications, and also basic principles of statistical inference; b)

be equipped with the necessary tools for statistical analysis of real time data

Unit – 1 [7 Hours]: Probability Theory-Sample Space, Events, Axioms of Probability, Other Definitions, Probability Rules, Conditional Probabilities, Bayes’ Theorem, Independence of Events;

Unit – 2 [12 Hours]: Random Variable and Distribution -Cumulative Distribution Function, Expectation, Variance, Moments, Skewness, Kurtosis, Quantiles, Median, Discrete Random Variables and Their Distributions, Continuous Random Variables and Their Distributions, Properties of PDF and PMF, Moment Generating Functions, Characteristic Function;

Unit – 3 [11 Hours]: Jointly Distributed Random Variable-Conditional Distributions: Discrete and Continuous case, Joint and Marginal Distribution, Covariance and

Correlation, Bivariate and Multivariate Normal Distribution, Functions of random variables and random vectors, distributions of sums of random variables;

Unit – 4 [5 Hours]: Limit Theorems and Inequalities-The Central Limit Theorem, Law of Large Numbers, Boole's Inequality, Bonferroni's Inequality, Chebyshev's and Markov's Inequality, Cauchy-Schwartz inequality, Jensen's Inequality;

Unit – 5 [6 Hours]: Statistical Inference-I-Distributions of the sample mean and the sample variance for a normal population, Chi-Square, t and F distributions. Estimation, Unbiasedness, consistency, the method of moments and the method of maximum likelihood estimation;

Unit – 6 [7 Hours]: Statistical Inference-II-Confidence intervals, Test of Hypothesis, Null and alternative hypotheses, the critical and acceptance regions, two types of error, power of the test. Z-test, t-test, Chi-square test, F-test. Computation, Simulation and Visualization using R or Matlab;

Text Books:

a) V K Rohatgi & A K M E Saleh, An Introduction to Probability and Statistics, 2nd Edition, Wiley, ISBN:9788126519262

b) S. Ross, A first Course in Probability, 8th Edition, Pearson, ISBN: 9780136033134 c) D.C. Montgomery, G.C. Runger, Applied Statistics and Probability for Engineers, 7th Edition, Wiley , ISBN:9781119456261

d) Michael J. Evans and Jefrey S. Rosenthal, Probability and Statistics, 2nd Edition, ISBN: 9781429224628

Reference Books:

a) Anderson, T.W., (1958) Introduction to Multivariate Statistical Analysis, Wiley: New York, ISBN:9788126524488

b) A.M. Gun, M.K. Gupta & B. Dasgupta, An Outline of Statistical Theory, Volume One, World Press

Homeworks & Scheduled Quizzes

In-class examinations