Probability and Statistics

These are self-study video lectures suitable for students studying college level mathematics, physics, economic, and social sciences. Only basic calculus is assumed. An Urdu version is available as well.

Lecture 1: 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 example.

Lecture 2: Factorials and their applications; Pascal’s Triangle; Counting outcomes; Conditional probability; Worked examples of conditional probability; Probability trees; Bayes’ Rule.

Lecture 3: Random variables; Geometric random variable; Binomial random variable; Expectation (mean) values; Worked examples; Poisson random variable; Call centre problem.

Lecture 4: 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.

Lecture 5: Covariance and correlation; Some simple properties of covariance; A physical understanding of covariance; Several discrete random variables; Illustrative solved examples.

Lecture 6: 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.

Lecture 7: 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).

Lecture 8: 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.

Lecture 9: One-dim drunkard’s walk – a first look; Random walk definitions; First return theorem; The eventual return problem; Random walk in two dimensions.

Lecture 10: States; Transitions; Markov Chains; Three examples of Markovian processes; Markovian basics; Regular (ergodic) Markov Chains; Absorbing Markov Chains.