Probability

By Dr. Sultan Sial

About this Course

 This is a first course in probability which provides basic concepts related to modelling of chance events in practical life. It provides preparation for further courses in stochastic processes, statistics, statistical mechanics and an understanding of the probability concepts essential for students who want to pursue studies in physical sciences, social sciences, economics, and engineering. The course starts with an introduction of probability terms and methods of computing simple and conditional probabilities. The concepts of discrete and continuous random variables are covered. Special discrete and continuous probability distributions are explored with their real life applications. 

What Will You Learn

Having successfully completed the course the students will be able to:

  • Explain the concepts of probability, including conditional probability 
  • Explain the concepts of random variable, probability distribution, distribution function
  • Explain the concepts of expected value, variance and higher moments, and calculate expected values and probabilities associated with the distributions of random variables 
  • Define and apply basic discrete and continuous distributions
  • Understand & apply Central Limit Theorem in different problems

Skills You Will Gain

  • Probability fundamentals
  • Random variables & distributions
  • Statistical thinking
  • Modeling uncertainty
  • Data & chance analysis

FORMAT

Free Open Courseware

LANGUAGE

Bilingual

DURATION

23 Lectures

QUANTITY

1

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Installment payment plans available

What is OpenCourseWare?

Access free online teaching material and resources from LUMS’ on-campus course offerings. Benefit from free access to cutting-edge research from our Schools and Centers.

How Will You Learn?

1

Explore our Course

Open Courseware are free courses recorded in LUMS

2

View our Playlist

Go through our YouTube Playlist with all the video lectures available

3

Learn at Your Own Pace

Complete the lectures at your own pace and time

4

Share your Learnings

Practice what you learn in real life and upskill yourself

Our Instructor(s)

Dr. Sultan Sial

Associate Professor,
Syed Babar Ali School of Science and Engineering, LUMS

Dr Sultan Sial received the MSc. Mathematics degree from Carleton University and the PhD. degree in Applied Mathematics from University of Western Ontario, Canada in 1992 and 1997. Prior to joining LUMS, he has been associated with University of Toronto, University of Western Ontario, Trent University, and Los Alamos National Lab (LANL). Dr Sial also has corporate sector experience; he has been the Vice President (Research) of Heuchera Technologies, and Vogelfrei Analytics. He has several publications, a book and book chapter in leading international journals.

Learn more

Dr. Sultan Sial

Associate Professor,
Syed Babar Ali School of Science and Engineering, LUMS

Dr Sultan Sial received the MSc. Mathematics degree from Carleton University and the PhD. degree in Applied Mathematics from University of Western Ontario, Canada in 1992 and 1997. Prior to joining LUMS, he has been associated with University of Toronto, University of Western Ontario, Trent University, and Los Alamos National Lab (LANL). Dr Sial also has corporate sector experience; he has been the Vice President (Research) of Heuchera Technologies, and Vogelfrei Analytics. He has several publications, a book and book chapter in leading international journals.

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Dr. Sultan Sial

Associate Professor,
Syed Babar Ali School of Science and Engineering, LUMS

Courses Taught

Dr. Sultan Sial

Associate Professor,
Syed Babar Ali School of Science and Engineering, LUMS

Dr Sultan Sial received the MSc. Mathematics degree from Carleton University and the PhD. degree…

Dr Sultan Sial received the MSc. Mathematics degree from Carleton University and the PhD. degree in Applied Mathematics from University of Western Ontario, Canada in 1992 and 1997. Prior to joining LUMS, he has been associated with University of Toronto, University of Western Ontario, Trent University, and Los Alamos National Lab (LANL). Dr Sial also has corporate sector experience; he has been the Vice President (Research) of Heuchera Technologies, and Vogelfrei Analytics. He has several publications, a book and book chapter in leading international journals.

Courses Taught

Teaching English

Dr. Sultan Sial

Associate Professor,
Syed Babar Ali School of Science and Engineering, LUMS

Dr Sultan Sial received the MSc. Mathematics degree from Carleton University and the PhD. degree…

Dr Sultan Sial received the MSc. Mathematics degree from Carleton University and the PhD. degree in Applied Mathematics from University of Western Ontario, Canada in 1992 and 1997. Prior to joining LUMS, he has been associated with University of Toronto, University of Western Ontario, Trent University, and Los Alamos National Lab (LANL). Dr Sial also has corporate sector experience; he has been the Vice President (Research) of Heuchera Technologies, and Vogelfrei Analytics. He has several publications, a book and book chapter in leading international journals.

Guest Instructors

Course Outline

In this module, you will learn about basic set-theoretic concepts that form the mathematical foundation of probability.

Sessions: 

  • Session 01: Overview of Set Theory

In this module, you will learn about counting principles, permutations, and combinations used to enumerate possible outcomes.

Sessions: 

  • Session 02: Counting
  • Session 03: Counting Exercises 

In this module, you will learn how to define sample spaces, events, and probability axioms for computing event probabilities.

Sessions: 

  • Session 04: Probability of Events

In this module, you will learn about conditional probability, total probability, and Bayes’ theorem for updating probabilities.

Sessions: 

  • Session 05: Conditional Probability
  • Session 06: Conditional Probability Examples
  • Session 07: Total Probability
  • Session 08: Bayes’ Theorem

In this module, you will learn about random variables and their probability distributions, covering both discrete and continuous models commonly used in practice. The module explores expectation, variance, and functions of random variables, along with important distributions such as Bernoulli, Poisson, Normal, Exponential, and Log-normal, emphasizing real-world applications.

Sessions: 

  • Session 09: Random Variables
  • Session 10: Functions of Random Variables, Expectation, Variance, Chebyshev’s Theorem 
  • Session 11: Uniform and Bernoulli Random Variables 
  • Session 12: Negative Binomial, Geometric, Multinomial Random Variables
  • Session 13: Hypergeometric Random Variable 
  • Session 14: Poisson and Exponential Random Variables 
  • Session 15: Normal Distribution 
  • Session 16: Log Normal Random Variable
  • Session 17: Maximum Likelihood Estimation 

In this module, you will learn about joint distributions and conditional distributions of multiple random variables.

Sessions: 

  • Session 18: Joint Probability Mass Functions/Joint Probability Density Functions
  • Session 19: Joint Distributions Continued
  • Session 20: Conditional Probability Mass Functions 
  • Session 21: Cumulative Conditional Mass Functions and Density Functions

In this module, you will learn about expectation, variance, covariance, correlation, moment generating functions, and probability inequalities.

Sessions: 

  • Session 22: Expectation, Variance, Covariance, Correlation 
  • Session 23: Moment Generating Functions 
  • Session 24: Markov and Chebyshev Inequalities 

In this module, you will learn about fundamental results describing long-run behavior of random variables and averages.

Sessions: 

  • Session 25: Law of Large Numbers
  • Session 26: Central Limit Theorem 

In this module, you will learn about discrete-time Markov chains and stochastic processes with memoryless behavior.

Sessions: 

  • Session 27: Discrete Time Markov Chains 

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