# Probability and Statistics

By Prof. Somesh Kumar   |   IIT Kharagpur
Learners enrolled: 4006
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 undergraduate
INDUSTRY 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.
Summary
 Course Status : Completed Course Type : Core Duration : 12 weeks Category : Mathematics Credit Points : 3 Level : Undergraduate/Postgraduate Start Date : 26 Jul 2021 End Date : 15 Oct 2021 Enrollment Ends : 09 Aug 2021 Exam Date : 23 Oct 2021 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

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
41. Additive Properties of Distributions-I
42. Additive Properties of Distributions-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

### Books and references

1. An Introduction to Probability and Statistics by V.K. Rohatgi & A.K. Md. E. Saleh
2. Probability and Statistical Inference by Hogg, R. V., Tanis, E. A. & Zimmerman D. L.
3. Probability and Statistics in Engineering by W.W. Hines, D.C. Montgomery, D.M. Goldsman, C.M. Borror
4. Introduction to Probability and Statistics for Engineers and Scientists by S.M. Ross
5. Introduction to Probability and Statistics by J.S. Milton & J.C. Arnold.
6. Introduction to Probability Theory and Statistical Inference by H.J. Larson
7. Probability and Statistics for Engineers and Scientists by R.E. Walpole, R.H. Myers, S.L. Myers, Keying Ye
8. Modern Mathematical Statistics by E.J. Dudewicz & S.N. Mishra
9. Introduction to the Theory of Statistics by A.M. Mood, F.A. Graybill and D.C. Boes

### Prof. Somesh Kumar

IIT Kharagpur
Prof. Somesh Kumar is a professor in the Department of Mathematics, IIT Kharagpur. He has over 32 years of experience of teaching courses on Probability Statistics, Statistical Inference, Sampling Theory, Stochastic Processes, Multivariate Analysis, Regression Analysis, Time Series, Experimental Designs, Decision Theory to undergraduate, postgraduate and doctorate students. His NPTEL courses (under MHRD) on Probability and Statistics, Statistical Inference and Statistical Methods for Scientists and Engineers (each of 40 hours) are available online and very popular. He has also taught Mathematics-I in QEEE program of MHRD to 130 engineering college students in online mode during Autumn 2014-2015. He offered the course “Probability and Statistics” for certification program in Jan-April 2016, Jan-April 2017, Jan-April 2019. He also offered the course “Statistical Inference” for certification program during Jan-April 2019. His lectures on “Probability” and “Permutation and combinations” for class XII students under IIT-PAL scheme of MHRD are also available through DTH channels of national television.

His research interests are Statistical Decision Theory, Estimation Theory, Testing of Hypothesis, Classification Problems, Directional Distributions, Limit Theorems. He has published more than 100 research papers in refereed reputed international journals and book chapters. He has supervised eleven Ph.D. students and more than two hundred fifty Masters (M.Tech./ M.Sc./B.Tech.) dissertations.

He has been guest professor in University of Ulm, Germany in July 2017 and June-July 2018 and in University of Dortmund in May-June 2019. He is Principal Investigator for a major research project “Drone for Vaccine Delivery” funded by the Indian Council for Medical Research. He has delivered invited lectures in various universities in India and abroad.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 23 October 2021 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kharagpur. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team