# Office of Academic AffairsIndian Institute of Science Education and Research Berhampur

Mathematics

MTH 414: Probability and Statistics (4)

Course contents:

Algebra of Sets: sets, classes, limit of a sequence of sets, rings, sigma rings, fields, sigma-fields, monotone classes.

Probability: Classical, relative frequency and axiomatic definitions of probability, addition rule and conditional probability, multiplication rule, total probability, Bayes Theorem and independence, problems.

Random Variables: Discrete, continuous and mixed random variables, probability mass, probability density and cumulative distribution functions, mathematical expectation, moments, probability and moment generating function, median and quantiles, Markov inequality, Chebyshevâ€™s inequality, problems.

Special Distributions, Joint Distributions: Joint, marginal and conditional distributions, product moments, correlation and regression, independence of random variables, bivariate normal distribution.

Transformations: functions of random vectors, distributions of order statistics, distributions of sums of random variables.

Sampling Distributions: The Central Limit Theorem, distributions of the sample mean and the sample variance for a normal population, Chi-Square, t and F distributions.

Descriptive Statistics: Graphical representation, Summarization and tabulation of data.

Estimation: Unbiasedness, consistency, the method of moments and the method of maximum likelihood estimation, confidence intervals for parameters in one sample and two sample problems of normal populations, confidence intervals for proportions.

Testing of Hypotheses: Null and alternative hypotheses, the critical and acceptance regions, two types of error, power of the test, the most powerful test and Neyman-Pearson Fundamental Lemma, tests for one sample and two sample problems for normal populations, tests for proportions, Chi-square goodness of fit test and its applications.

Suggested Books:

• W. Feller, An Introduction to Probability Theory and Its Applications, Volume 1, 3rd Edition, Wiley, 1968
• V. Rohatgi, A. Saleh, Introduction to Probability Theory and Statistics, 2nd Edition, Wiley, 2000
• S.M. Ross, A First Course in Probability, 6th Edition, Prentice Hall
• A. Craig, R. Hogg, J. McKean, Introduction to Mathematical Statistics, 6th Edition, Prentice Hall, 2004
• J.S. Milton and J.C. Arnold, Introduction to Probability and Statistics
• P. Hoel, S. Port, C. Stone, Introduction to Probability Theory, 1st Edition, Brooks Cole, 1972

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