**DESCRIPTION OF COURSES**

**AS 560 PROBABILITY THEORY (2L+0P) I**

**Objective**

This is a fundamental course in Statistics. This course lays the foundation of probability theory, random variable, probability distribution, mathematical expectation, etc. which forms the basis of basic statistics. The students are also exposed to law of large numbers and central limit theorem. The students also get introduced to stochastic processes.

**Theory**

UNIT I

Elements of measure theory. Probability - classical and frequency definitions, axiomatic approach, laws of probability, conditional probability, Bayes theorem, Class of sets, field, sigma field, minimal sigma field, Borel sigma field in R.

UNIT II

Random variable- discrete and continuous. Probability mass and probability density functions, distribution function. Mathematical expectation and its laws. Probability generating, moment generating and characteristic functions. Inversion and Uniqueness theorems for characteristic functions. Raw and central moments and their relation.

UNIT III

Markov’s, Chebychev’s and Kolmogorov’s inequalities. Modes of stochastic convergence. Jenson, Liapounov, holder’s and Minkowsky’s inequalities. Sequence of random variables and modes of convergence (convergence in distribution, in probability, almost surely, and quadratic mean) and their interrelations. Statement of Slutsky’s theorem. Borel –Cantelli lemma and Borel 0-1 law.

UNIT IV

Weak and strong laws of large numbers, Central limit theorems (CLT). Demoviere- Laplace CLT, Lindberg – Levy CLT, Liapounov CLT, Statement of Lindeberg-Feller CLT and simple applications. Definition of quantiles and statement of asymptotic distribution of sample quantiles.

**Suggested Readings**

- Ash, Robert B. 2000.
*Probability and Measure Theory*. Academic Press. - Bhat, B.R. 1989.
*Modern Probability Theory*. Wiley Eastern Private Ltd. - Billingsley, P. 1986.
*Probability and Measure*. John Wiley. - Capinski, M. and Zastawniah. 2001.
*Probability Through Problems*. Springer. - Dudewicz, E.J. and Mishra, S.N. 1988.
*Modern Mathematical Statistics*. John Wiley. - Emanuel Parzen. 1960.
*Modern Probability Theory and its Application*. Wiley Eastern Private Ltd. - Enders, A.R.1985.
*Probability Theory and Application*. Inter national Human Resources Development Corporation. - Feller, W. 1970.
*An Introduction to Probability Theory and its Applications*. - John Wiley. Janos Galambos.1984.
*Introductory to Probability Theory*. Marcel Dekker. - Laha, R.G. and Rohatgi, V.K. 1979.
*Probability Theory*. John Wiley. - Larson, Harold J. 1969.
*Introduction to Probability Theory and Statistical Inference*, John Wiley. - Loeve, M. 1978.
*Probability Theory*. Springer. - Marek Fisz. 1963.
*Probability Theory and Mathematical Statistics*. John Wiley. - Pfeiffer, Paul E. 1990.
*Probability for Applications*. Springer Verlog. - Robert, B.A. 2000.
*Probability and Measure Theory*. Wiley Eastern Private Ltd. - Rohatgi, V.K. and Ehsan, S. 2003.
*An Introduction to Probability Theory and Mathematical Statistics*. Wiley Eastern Private Ltd. - Rohatgi, V.K. and Saleh, A.K. Md. E. 2005.
*An Introduction to Probability and Statistics.*John Wiley. - William Feller. 1972.
*An Introduction to Probability Theory and its Applications*. Wiley Eastern Private Ltd.