Probability Theory

Paper Code: 
24STT122
Credits: 
6
Contact Hours: 
90.00
Max. Marks: 
100.00
Objective: 

This is a fundamental course in Statistics. This course lays the foundation of probability theory, random variable, probability distribution, mathematical expectation, etc. which forms the basis of basic statistics. The students are also exposed to the law of large numbers.

 

Course Outcomes: 

Course

Course Outcomes

Learning and teaching strategies

Assessment Strategies

Course Code

Course Title

24STT122

Probability Theory

(Theory)

CO 7: Identify the problem and apply appropriate laws of probability and Bayes theorem.

CO 8: Apply the knowledge of distribution function, conditional probability and transformations on various distributions.

CO 9: Obtain the constants of population using expectation, moment generating function and cumulant generating functions.

CO 10: Solve the various laws of large numbers and inequalities to sequences of random variables.

CO 11: Formulate correlation and simple linear regression model to real life examples.

CO 12: Contribute effectively in course-specific interaction.

Approach in teaching: 

Interactive Lectures, 

Group Discussion, 

Classroom Assignment,

Problem Solving Sessions.

 

Learning activities for the students:

Assignments,

Seminar,

Presentation,

Subject based  Activities.

Classroom Quiz, Assignments, Class Test,

Individual Presentation.

 

18.00
Unit I: 
Basic Probability

General probability space, various definitions of probability, combinations of events, additive and multiplicative law of probability, conditional probability, Bayes’ theorem and its application.

 

18.00
Unit II: 
Random Variable

Concept of random variable, cumulative distribution function, probability distribution function, joint probability distribution function, marginal distribution function and their application, conditional distribution function and conditional probability distribution function of random variables and their distributions using: jacobian transformation, cumulative distribution function, moment generating function.

 

18.00
Unit III: 
Mathematical Expectation, Moments and Characteristic Function

Mathematical Expectation, moments, Sheppard’s correction, conditional expectation, moment generating function and their applications, cumulant generating function and their applications, characteristic function and its applications. Inversion Theorem, Continuity Theorem, Uniqueness Theorem.

 

 

18.00
Unit IV: 
Probability Inequalities and Convergence

Markov and Jenson and their applications, Chebychev inequality (without proof) with simple numerical, Convergence in probability and convergence in distribution, weak law of large numbers.

 

18.00
Unit V: 
Correlation and Regression

Introduction to correlation and its types, Measures of correlation coefficient, multiple and partial correlation, intra class correlation and correlation ratio. Method of least square for linear regression (one independent variable).

 

Essential Readings: 
  • Kingman, J.F. & Taylor, S.J. (1996): Introduction to Measure and Probability, Cambridge Univ. Press.
  • Loeve (1996): Probability Theory, Affiliated East –West Press Pvt. Ltd. New Delhi.
  • Bhatt, B.R. (2000): Probability, New Age International India.
  • Feller,W.(1971): Introduction to Probability Theory and its Applications, Vol. I and II. Wiley, Eastern-Ltd.
  • Rohatgi, V.K (1984): An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern, third edition.

 

SUGGESTED READINGS:

  • Billingsley, P. (1986): Probability and Measure, John Wiley Publications, fourth edition.
  • Dudley, R.M. (1989): Real Analysis and Probability, Worlds Worth & Books.
  • Tucket H.G. (1967): A Graduate Course in Probability, Academic Press.
  • Basu, A.K. (1999): Measure Theory and Probability, PHI.

 

e-RESOURCES:

 

JOURNALS:

  • Sankhya The Indian Journal of Statistics, Indian Statistical Institute
  • Aligarh Journal of Statistics, Department of Statistics and Operations Research, Aligarh Muslim University
  • Afrika Statistika, Saint-Louis Senega University
  • International Journal of Statistics and Reliability Engineering, Indian Association for Reliability and Statistic
  • Journal of the Indian Society for Probability and Statistics, Indian Society for Probability and Statistics
  • Journal of the Indian Statistical Association, Indian Statistical Association
  • Statistica, Department of Statistical Sciences Paolo Fortunato, University of Bologna
  • Statistics and Applications, Society of Statistics, Computer and Applications
  • Stochastic Modeling and Applications, MUK Publications and Distributions

 

Academic Year: 
2024-2025
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