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# Introduction to Probability Theory and Stochastic Processes (Tamil)

### Overview

ABOUT THE COURSE: This course explanations and expositions of probability and stochastic processes concepts which they need for their experiments and research. It also covers theoretical concepts of probability and stochastic processes pertaining to handling various stochastic modeling. This course provides random variable, distributions, moments, modes of convergences, classification and properties of stochastic processes, stationary processes, discrete and continuous time Markov chains and simple Markovian queueing models.INTENDED AUDIENCE: UG and research scholarsPREREQUISITES: Calculus or Mathematics 1INDUSTRY SUPPORT: Goldman Sachs, RBS, all financial companies

### Syllabus

Week 1:Basics of Probability
1. Random experiment, sample, event, axioms of probability, probability space
2. Conditional probability, independence of events
3. Total probability rule, multiplication rule, Baye's theorem.
Week 2:Random Variable
1. Definition, cumulative distribution function,
2. Type of random variables, probability mass function, probability density function
3. Distribution of function of random variable.
Week 3:Moments and Inequalities
1. Mean and variance
2. Higher order moments, moments inequalities
3. Generating functions.
Week 4:Standard Distributions
1. Some common discrete distributions
2. Some common continuous distributions.
3. Some applications of random variable
Week 5:Higher Dimensional Distributions
1. Two and higher dimensional distributions, joint distributions
2. Joint probability mass function, joint probability density function
3. Independent random variables
Week 6:Functions of Several Random Variables
1. functions of several random variables
2. order statistics
3. Conditional distributions, random sum.
Week 7:Cross Moments
1. Moments of functions of several random variables, Covariance-variance matrix
2. Correlation coefficient, linear regression
3. Conditional expectation.
Week 8:Limiting Distributions
1. Modes of convergences
2. Law of large numbers
3. Central limit theorem.
Week 9:Introduction to Stochastic Processes (SPs)
1. Definition and examples of SPs, Classification of random processes according to state space and parameter space
2. Some common stochastic processes, examples
3. Weakly stationary and strongly stationary processes, Moving average and auto regressive processes, examples.
Week 10:Discrete-time Markov Chains (DTMCs)
1. Definition and examples of DTMC, transition probability matrix, Chapman-Kolmogorov equations
2. Calculation of n-step transition probabilities, limiting probabilities, classification of states ergodicity
3. Stationary distribution, random walk and gambler’s ruin problem.
Week 11:Continuous-time Markov Chains (CTMCs)
1. Definition and examples of CTMC, Kolmogorov equations, infinitesimal generator
2. Definition of Birth death processes, examples, Pure birth processes, pure death processes, Poisson process
3. Steady state probabilities, Time-dependent probabilities.
Week 12:Simple Markovian Queueing Models
1. M/M/1, M/M/1/N
2. M/M/c/N, M/M/N/N, steady state probabilities,
3. Some important measures, examples.

### Taught by

Prof. S Dharmaraja

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