MTL 601 (Probability and Statistics)

4 Credits (3-1-0)

I Semester 2023 - 2024

Lectures: Monday and Wednesday between 11:00 AM and 11:50 AM and Thursday between 12:00 noon and 12:50 PM.

INFORMATION SHEET

Axiomatic definition of a probability measure, examples, properties of the probability measure, finite probability space, conditional probability and Baye's formula, countable probability space, general probability space

Random variables, examples, sigma-field generated by a random variable, tail sigma-field, probability space on R induced by a random variable

Distribution - definition and examples, properties, characterization, Jordan decomposition theorem, discrete, continuous and mixed random variables, standard discrete and continuous distributions

Two dimension random variables, joint distributions, marginal distributions, operations on random variables and their corresponding distributions, multidimensional random variables and their distributions

Expectation of a random variable, expectation of a discrete and a continuous random variable, moments and moment generating function, correlation, covariance and regression

Various modes of convergence, convergence in distribution, weak convergence of generalized distributions, Helly-Bray theorems, Scheffe's theorem

Characteristic function – definition and examples, properties, conjugate distributions, uniqueness and inversion theorems, moments using characteristic function, Paul Levy's continuity property of characteristic functions

Independent events, sigma-fields and random variables, characterization of independent random variables, Borel 0-1 criteria, Kolmogorov 0-1 criteria

Weak law of large numbers, strong law of large numbers, central limit theorem – Liapunov's and Lindberg's condition, Lindeberg-Levy form

Sampling distributions, characteristics, asymptotic properties

Theory of estimation – Classification of estimates, methods of estimates, confidence regions, MVUE, Cramer Rao Theorem, Rao Blackwellization

Tests of significance – General theory of testing hypothesis, choice of a test, simple and composite hypothesis, tests of simple and composite hypothesis

Goodness of fit test, Chi-square test, Kolmogorov Smirnov test, analysis of variance

Main Text Books

1.    An Introduction to Probability and Statistics, Vijay K. Rohatgi and A.K. Md. Ehsanes Saleh, John Wiley, second edition, 2001.

2.    Introduction to Probability and Stochastic Processes with Applications, Liliana Blanco Castaneda, Viswanathan Arunachalam, Selvamuthu Dharmaraja, Wiley, Asian Edition, Jan. 2016.

3.    Introduction to Statistical Methods, Design of Experiments and Statistical Quality Control, Selvamuthu Dharmaraja, Dipayan Das, Springer, 2018.

Reference Text Books

1.    Introductory Probability and Statistical Applications, Paul L. Mayer, Addison-Wesley, Second Edition, 1970.

2.    Statistical Inference, George Casella and Roger L. Beger Saleh, Duxbury Press, second edition, 2001.

Lecture Notes
Introduction to this course

Sl. No. Topics Videos Lecture Class Notes

1 Axioms of probability, Probability space, Conditional probability, Independence, Baye's rule Part 1, Part 2, Part 3, Part 4, Part 5 1

2 Random variable, Distribution of Functions of Random Variable, Moments, Generating functions Part 1, Part 2, Part 3, Part 4, Part 5, Part 1, Part 2, Part 3, Part 4, Part 5 2

3 Some common discrete and continuous distributions Part 6, Part 7, Part 8, Part 9, Part 10, Part 11 3

4 Two and higher dimensional distributions, Functions of random variables, Order statistics, Conditional distributions, Covariance, correlation coefficient, conditional expectation Part 1, Part 2, Part 3, Part 4, Part 5, Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14 4

5 Modes of convergences, Law of large numbers, Central limit theorem Part 1, Part 2, Part 3, Part 4, Part 5 5

6 Descriptive Statistics Part 1 6

7 Sampling Distributions Part 1, Part 2 7

8 Point and Interval Estimations Part 1, Part 2, Part 3 8

9 Hypothesis Testing Part 1, Part 2, Part 3 9

10 Analysis of Correlation and Regression Part 1, Part 2, Part 3 10

Note: The above classification of lecture notes are tentative only. (courtesy: NPTEL course)

Sl. No.	Topics	Videos	Lecture Class Notes
1	Axioms of probability, Probability space, Conditional probability, Independence, Baye's rule	Part 1, Part 2, Part 3, Part 4, Part 5	1
2	Random variable, Distribution of Functions of Random Variable, Moments, Generating functions	Part 1, Part 2, Part 3, Part 4, Part 5, Part 1, Part 2, Part 3, Part 4, Part 5	2
3	Some common discrete and continuous distributions	Part 6, Part 7, Part 8, Part 9, Part 10, Part 11	3
4	Two and higher dimensional distributions, Functions of random variables, Order statistics, Conditional distributions, Covariance, correlation coefficient, conditional expectation	Part 1, Part 2, Part 3, Part 4, Part 5, Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14	4
5	Modes of convergences, Law of large numbers, Central limit theorem	Part 1, Part 2, Part 3, Part 4, Part 5	5
6	Descriptive Statistics	Part 1	6
7	Sampling Distributions	Part 1, Part 2	7
8	Point and Interval Estimations	Part 1, Part 2, Part 3	8
9	Hypothesis Testing	Part 1, Part 2, Part 3	9
10	Analysis of Correlation and Regression	Part 1, Part 2, Part 3	10