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# Probability and Statistics

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## Overview

The use of statistical reasoning and methodology is indispensable in modern world. It is true for any discipline, be it physical sciences, engineering and technology, economics or social sciences. Much of the advanced research in biology, genetics, and information science relies increasingly on use of statistical tools. It is essential for the students to get acquainted with the subject of probability and statistics at an early stage. The present course has been designed to introduce the subject to undergraduate/postgraduate students in science and engineering. The course contains a good introduction to each topic and an advance treatment of theory at a fairly understandable level to the students at this stage. Each concept has been explained through examples and application oriented problems.

## Syllabus

Week 1: Sets, Classes, Collections

Sequence of Sets

Ring, Field (Algebra)

Sigma-Ring, Sigma-Field, Monotone Class

Random Experiment, Events

Definitions of Probability

Properties of Probability Function-I

Properties of Probability Function-II

Week 2: Conditional Probability

Independence of Events

Problems in Probability-I

Problems in Probability-II

Random Variables

Probability Distribution of a Random Variable-I

Week 3: Probability Distribution of a Random Variable-II

Moments

Characteristics of Distributions-I

Characteristics of Distributions-II

Special Discrete Distributions-I

Special Discrete Distributions-II

Special Discrete Distributions-III

Week 4: Poisson Process-I

Poisson Process-II

Special Continuous Distributions-I

Special Continuous Distributions-II

Special Continuous Distributions-III

Special Continuous Distributions-IV

Special Continuous Distributions-V

Week 5: Normal Distribution

Problems on Normal Distribution

Problems on Special Distributions-I

Problems on Special Distributions-II

Function of a Random Variable-I

Function of a Random Variable-II

Week 6: Joint Distributions-I

Joint Distributions-II

Independence, Product Moments

Linearity Property of Correlation and Examples

Bivariate Normal Distribution-I

Bivariate Normal Distribution-II

Week 7: Additive Properties of Distributions-I

Transformation of Random Variables

Distribution of Order Statistics

Basic Concepts

Chi-Square Distribution

Week 8: Chi-Square Distribution
(Cont…), t-Distribution

F-Distribution

Descriptive Statistics – I

Descriptive Statistics - II

Descriptive Statistics – III

Descriptive Statistics – IV

Week 9: Introduction to Estimation

Unbiased and Consistent Estimators

LSE, MME

Examples on MME, MLE

Examples on MLE-I

Examples on MLE-II, MSE

Week 10: UMVUE, Sufficiency, Completeness

Rao-Blackwell Theorem and its Applications

Confidence Intervals-I

Confidence Intervals- II

Confidence Intervals- III

Confidence Intervals- IV

Week 11: Basic Definitions

Two Types of Errors

Neyman-Pearson Fundamental Lemma

Applications of N-P Lemma-I

Applications of N-P Lemma-II

Week 12: Testing for Normal Mean

Testing for Normal Variance

Large Sample Test for Variance and Two Sample Problem

Paired t-Test

Examples

Week 13: Testing Equality of Proportions

Chi-Square Test for Goodness Fit -I

Chi-Square Test for Goodness Fit –II

Testing for Independence in  Contingency Table –I

Testing for Independence in  Contingency Table-II

Somesh Kumar

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