Upcoming conference symposium workshop

Brief description about the school: Probability theory is now considered an integral part of mathematics. The mathematics programmes of most Indian Universities, however, do not include probability theory in much detail. The aim of this school is to give a comprehensive training in probability and stochastic processes to students of the undergraduate and postgraduate programmes. Besides, giving them a chance to interact with researchers in these topics is also a goal of this instructional school.
Three courses -- 1) Measure theoretic probability, 2) Conditional probability and martingales , 3) Brownian motion will be taught here.

Who can Apply?

Any Master degree students and PhD students can apply.

To Apply:

Please go to NCM, Mumbai website and complete the application process by registering there.

Last date for receiving online applications: 14th May 2023 (Link for application will be provided here shortly.)

Instructors:

• Anish Sarkar, ISI, Delhi
• Soumendu Sudar Mukherjee, ISI, Kolkata
• Arijit Chakrabarty, ISI, Kolkata
• Manjunath Krishnapur, IISc, Bengaluru
• Suprio Bhar, IIT, Kanpur
• Arup Bose, ISI, Kolkata

Detailed Syllabus:

Measure theoretic Probability: Caratheodory extension theorem , Monotone class theorem, Dynkin’s pi-lambda theorem, MCT, Fatou’s Lemma, DCT, Fubini’s theorem. Probability spaces, random variables and random vectors, expected value and its properties. Independence. Various modes of convergence and their relation. The Borel-Cantelli lemmas. Weak Law of large numbers for i.i.d. finite mean case. Kolmogorov 0-1 law, Kolmogorov’s maximal inequality. Statement of Kolmogorov’s three-Series theorem (proof if time permits). Strong law of large numbers for i.i.d. case. Characteristic functions and its basic properties, inversion formula, Levy’s continuity theorem. Lindeberg CLT, CLT for i.i.d. finite variance case, Lyapunov CLT.

Conditional probability and martingales: Absolute continuity and singularity of measures. Hahn-Jordon decomposition, Radon-Nikodym Theorem, Lebesgue decomposition. Conditional expectation – Definition and Properties. Regular conditional probability, proper RCP. Regular conditional distribution. Discrete parameter martingales, sub-and super-martingales. Doob’s Maximal Inequality, Upcrossing inequality, martingale convergence theorem, Lp inequality, uniformly integrable martingales, reverse martingales, Levy’s upward and downward theorems. Stopping times, Doob’s optional sampling theorem. Discrete martingale transform, Doob’s Decomposition Theorem. Applications of martingale theory: SLLN for i.i.d. random variables.

Brownian Motion: Introduction to Brownian Motion, Kolmogorov Consistency theorem, Kolmogorov Continuity theorem, Construction of BM. Basic Martingale Properties and path properties – including Holder continuity and non-differentiability. Quadratic variation. Markov Property and strong Markov property of BM, reflection principle, Blumenthal’s 0-1 law. Distributions of first passage time and of running maximum of BM.

Brief description about the school:

Under choice based credit system, UGC made it compulsory to teach ‘Probability and Statistics’ for each Mathematics (Honours) students. Since most of the university curriculum does not have probability and statistics, college teachers are not competent enough to teach that course properly. This school will try to bridge that gap, by exposing the college teachers with the experts of these fields to enlighten their viewpoint of these subjects while teaching. We run two courses -- 1) Probability, 2) Statistics in this program.

Who can Apply?

Only NET/SET qualified College or University teachers can apply here. Preference will be given to young teachers who are pursuing a doctoral degree.

To apply: 

Please go to NCM, Mumbai website and complete the application process by registering there.

Last date for receiving online applications: 28th May 2023 

Instructors:

• Alok Goswami, Indian Association for the Cultivation of Science, Kolkata
• Arnab Chakraborty, ISI, Kolkata
• B V Rao, CMI, Chennai
• Mousumi Bose, ISI, Kolkata

Syllabus:

• Probability: Random experiment, probability space, finite sample spaces. Conditional probability, Bayes theorem, independence. Real random variables (discrete and continuous), cumulative distribution function, probability mass/density functions, mathematical expectation, moments, moment generating function, characteristic function. Discrete distributions : uniform, binomial, Poisson, geometric, negative binomial, Continuous distributions : uniform, normal, exponential. Joint cumulative distribution function and its properties, joint probability density functions, marginal and conditional distributions, expectation of function of two random variables, moments, covariance, correlation coefficient, independent random variables, joint moment generating function (jmgf) and calculation of covariance from jmgf, characteristic function. Conditional expectations, linear regression for two variables, regression curves. Bivariate normal distribution. Markov and Chebyshev’s inequality, Convergence in Probability, statement and interpretation of weak law of large numbers and strong law of large numbers. Central limit theorem for independent and identically distributed random variables with finite variance.
• Statistics: Sampling and Sampling Distributions : Populations and Samples, Random Sample, distribution of the sample, Simple random sampling with and without replacement. Sample characteristics. Sampling Distributions : Statictic, Sample moments. Sample variance, Sampling from the normal distributions, Chi-square, t and F-distributions, Estimation of parameters : Point estimation. Interval Estimation- Confidence Intervals for mean and variance of Normal Population. Mean-squared error. Properties of good estimators - unbiasedness, consistency, sufficiency, Minimum-Variance Unbiased Estimator (MVUE). Method of Maximum likelihood : likelihood function, ML estimators for discrete and continuous models. Statistical hypothesis : Simple and composite hypotheses, null hypotheses, alternative hypotheses, one-sided and two-sided hypotheses. The critical region and test statistic, type I error and type II error, level of significance. Power function of a test, most powerful test. The p-value (observed level of significance), Calculating p-values. Simple hypothesis versus simple alternative : Neyman-Pearson lemma (Statement only). Bivariate frequency Distribution : Bivariate data, Scatter diagram, Correlation, Linear Regression, principle of least squares and fitting of polynomials and exponential curves.