DESCRIPTION OF COURSES
AS 560 PROBABILITY THEORY (2L+0P) I
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.
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.
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.
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.
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.