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# Probability and Statistics

## Overview

The use of statistical reasoning and methodology is indispensable in modern world. It is applicable to every discipline, be it physical sciences, engineering and technology, economics or social sciences. Much of the advanced research in electronics, electrical, computer science, industrial engineering, 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. INTENDED AUDIENCE : Any Interested Learners.PREREQUISITES : Must have good knowledge of Differential and Integral Calculus, sequences and series, Basic Linear/Matrix Algebra (usually students who have completed Mathematics-I and II at first year undergraduateINDUSTRY SUPPORT : Today all industries use statistical methods. So for students desirous to work in any type of industry, this course will be indispensable. In particular, companies dealing with Business Analytics, Banking and finance, Insurance machine learning, data mining etc. this course will be invaluable.

## Syllabus

Week 1:
1. Sets, Classes, Collections
2. Sequence of Sets
3. Ring, Field (Algebra)
4. Sigma-Ring, Sigma-Field, Monotone Class
5. Random Experiment, Events
6. Definitions of Probability
7. Properties of Probability Function-I
8. Properties of Probability Function-II
Week 2:
9. Conditional Probability
10. Independence of Events
11. Problems in Probability-I
12. Problems in Probability-II
13. Random Variables
14. Probability Distribution of a Random Variable-I
Week 3:
15. Probability Distribution of a Random Variable-II
16. Moments
17. Characteristics of Distributions-I
18. Characteristics of Distributions-II
19. Special Discrete Distributions-I
20. Special Discrete Distributions-II\
Week 4:
21. Special Discrete Distributions-III
22. Poisson Process-I
23. Poisson Process-II
24. Special Continuous Distributions-I
25. Special Continuous Distributions-II
26. Special Continuous Distributions-III
Week 5:
27. Special Continuous Distributions-IV
28. Special Continuous Distributions-V
29. Normal Distribution
30. Problems on Normal Distribution
31. Problems on Special Distributions-I
32. Problems on Special Distributions-II
Week 6:
33. Function of a Random Variable-I
34. Function of a Random Variable-II
35. Joint Distributions-I
36. Joint Distributions-II
37. Independence, Product Moments
38. Linearity Property of Correlation and Examples
Week 7:
39. Bivariate Normal Distribution-I
40. Bivariate Normal Distribution-II
43. Transformation of Random Variables
44. Distribution of Order Statistics
Week 8:
45. Basic Concepts
46. Chi-Square Distribution
47. Chi-Square Distribution (Cont…), t-Distribution
48. F-Distribution
49. Descriptive Statistics – I 50. Descriptive Statistics – II
Week 9:
51. Descriptive Statistics – III
52. Descriptive Statistics – IV
53. Introduction to Estimation
54. Unbiased and Consistent Estimators
55. LSE, MME 56. Examples on MME, MLE
Week 10:
57. Examples on MLE-I
58. Examples on MLE-II, MSE
59. UMVUE, Sufficiency, Completeness
60. Rao-Blackwell Theorem and its Applications
61. Confidence Intervals-I
62. Confidence Intervals- II 63. Confidence Intervals- III\
Week 11:
64. Confidence Intervals- IV
65. Basic Definitions
66. Two Types of Errors
67. Neyman-Pearson Fundamental Lemma 68. Applications of N-P Lemma-I
69. Applications of N-P Lemma-II
Week 12:
70. Testing for Normal Mean
71. Testing for Normal Variance
72. Large Sample Test for Variance and Two Sample Problem
73. Paired t-Test
74. Examples
75. Testing Equality of Proportions

Somesh Kumar

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