Eqbal Ahmad Centre for Public Education

Languages: Urdu, English

These are self-study lectures on probability theory suitable for college level students studying mathematics, physics, economics, and business.  

Module 1.1 (English, Urdu)

• Random experiments, sample space and events, A quick review of sets, Venn diagrams and consequences, Sample space and events — examples, Axioms of probability, Probability — elementary examples

Module 1.2 (English, Urdu)

• Factorials and their applications, Pascal’s Triangle, Counting outcomes, Conditional probability, Worked examples of conditional probability, Probability trees, Bayes’ Rule

Module 1.3 (English, Urdu)

• Random variables, Geometric random variable, Binomial random variable, Expectation (mean) values, Worked examples. Poisson random variable, Call centre problem

Module 1.4 (English, Urdu)

• Variance and standard variation, E[X] and E[X^2] for binomial distribution, The Chebyshev Rule, Cumulative Distribution Function, Continuous random variables, Connecting discrete to continuous probabilities, Examples of continuous probability distributions

Module 1.5 (English, Urdu)

• Covariance and Correlation, Some simple properties of covariance, A physical understanding of covariance, Several discrete random variables, Illustrative solved examples

Module 1.6 (English, Urdu)

• Two continuous random variables, Some basic formulae, Practice with double integrals, Example: trains arriving at a station, Example: aiming darts at a board, Example: Buffon’s needle

Module 1.7 (English, Urdu)

• A function of generating moments, Example: Binomial Distribution, Example: Poisson Distribution, Example: Exponential Distribution, Introduction to Normal Distribution, Binomial Distribution Normal Distribution (proof), Law of Large Numbers (with proof)

Module 1.8 (English, Urdu)

• Central Limit Theorem – with potatoes, CLT – why it is astonishing, Generating moments of the normal distribution, Generating moments for n independent variables, Proof of CLT, CLT for i.i.d’s that are uniformly distributed, CLT – three applications

Module 1.9 (English, Urdu)

• One-dim drunkard’s walk – a first look, Random walk definitions, First return theorem, The eventual return problem, Random walk in two dimensions

Module 1.10 (English, Urdu)

• States, Transitions, Markov Chains, Three examples of Markovian processes, Markovian basics, Regular (ergodic) Markov Chains, Absorbing Markov Chains

Recommended textbooks and materials:

1. A Natural Introduction to Probability Theory, Robert Meester, Birkhäuser Verlag (2008).

2. Introduction to Probability Models, 9th Edition, Elsevier (2009).

3. Grinstead and Snell’s Introduction to Probability, the CHANCE Project (2006).

4. Probability Theory and its Applications, William Feller, 3rd edition, John Wiley (1970).

For worked examples, I strongly recommend the books by Hossein Pishro-Nik which you can easily locate with a Google search.