Probability Mathematics Course

Lesson 1 • Axioms and Basic Properties

Presents Kolmogorov's three axioms and derives fundamental rules from them. Students use these rules to verify and compute probabilities rigorously.

Lesson 2 • Sets, Outcomes, and Sample Spaces

Introduces set notation and the concept of a sample space as the universe of outcomes. Establishes the vocabulary used throughout all probability calculations.

Lesson 3 • Defining and Measuring Probability

Covers the classical, frequentist, and subjective interpretations of probability. Connects each interpretation to real measurement contexts.

Lesson 1 • Multiplication Rule and Chain Rule

Derives the multiplication rule from the conditional probability definition. Extends it to chains of multiple dependent events.

Lesson 2 • Bayes' Theorem and Updating Beliefs

Derives Bayes' theorem and applies it to reverse conditional probabilities. Students update prior beliefs with new evidence in structured problems.

Lesson 3 • Law of Total Probability

Introduces partitions of the sample space and the total probability formula. Provides the foundation needed to derive Bayes' theorem.

Lesson 4 • Conditional Probability Defined

Defines P(A|B) as a ratio and explains how conditioning restricts the sample space. Builds the conceptual bridge from unconditional to conditional reasoning.

Lesson 1 • Expected Value and Variance

Derives expectation as a probability-weighted average and variance as spread. Establishes linearity of expectation and its computational power.

Lesson 2 • Poisson Distribution

Derives the Poisson distribution as a limit of the binomial for rare events. Students apply it to count data and verify the Poisson approximation conditions.

Lesson 3 • Geometric and Negative Binomial Distributions

Models waiting times until success using geometric and negative binomial distributions. Connects these to the memoryless property of geometric trials.

Lesson 4 • Bernoulli and Binomial Distributions

Models binary trials with Bernoulli variables and repeated trials with the binomial. Students derive and apply the PMF, mean, and variance formulas.

Lesson 1 • Probability Density Functions

Defines the PDF and explains why point probabilities are zero for continuous variables. Connects the PDF to the CDF through integration.

Lesson 2 • Uniform and Exponential Distributions

Introduces the uniform distribution for equally-likely intervals and the exponential for waiting times. Highlights the exponential's memoryless property.

Lesson 3 • Other Key Continuous Distributions

Surveys gamma, beta, and log-normal distributions and their parameter roles. Prepares students to recognise and select appropriate models for varied data shapes.

Lesson 4 • Expectation and Variance for Continuous Variables

Adapts expectation and variance formulas to integrals for continuous distributions. Reinforces moment calculations as a unified framework across variable types.

Lesson 1 • Conditional Distributions

Extends conditional probability to full conditional distributions of one variable given another. Connects to earlier Bayes' theorem concepts in a distributional framework.

Lesson 2 • Joint Probability Distributions

Defines joint PMFs and PDFs for pairs of random variables. Students extract marginal distributions by summing or integrating out one variable.

Lesson 3 • Transformations of Random Variables

Derives distributions of functions of one or more random variables. Covers the change-of-variables technique and the Jacobian for continuous cases.

Lesson 4 • Multivariate Normal Distribution

Introduces the bivariate and multivariate normal distribution and its covariance matrix. Students interpret marginal and conditional normals and their geometric meaning.

Lesson 1 • Modes of Convergence

Distinguishes convergence in probability, almost sure convergence, and convergence in distribution. Clarifies the hierarchy and relationships among these modes.

Lesson 2 • Moment-Generating Functions

Introduces MGFs as a tool for deriving moments and proving distributional results. Students use MGFs to establish the CLT and identify distributions.

Lesson 3 • Central Limit Theorem

Proves that standardised sample means converge to a normal distribution. Students apply the CLT to approximate probabilities for sums and averages.

Lesson 4 • Inequalities and Probability Bounds

Derives Markov's and Chebyshev's inequalities as tools for bounding tail probabilities. Provides rigorous bounds without requiring knowledge of the full distribution.

Lesson 1 • Confidence Intervals

Constructs confidence intervals for means and proportions using sampling distributions. Students interpret coverage probability and interval width trade-offs.

Lesson 2 • Common Parametric Tests

Applies the hypothesis testing framework to z-tests, t-tests, and chi-squared tests. Connects each test to its underlying probability model.

Lesson 3 • Sampling Distributions

Derives the distributions of sample statistics such as the mean and variance. Introduces chi-squared, t, and F distributions as tools for inference.

Lesson 4 • Point Estimation

Defines estimator properties including unbiasedness, consistency, and efficiency. Students apply method-of-moments and maximum likelihood estimation.

Lesson 1 • Queuing Theory Basics

Models waiting lines using M/M/1 and M/M/c queue structures. Students compute average queue length, wait time, and server utilisation.

Lesson 2 • Stationary Distributions and Long-Run Behaviour

Derives stationary distributions and conditions for their existence and uniqueness. Students interpret steady-state probabilities in applied settings.

Lesson 3 • Poisson Processes

Defines the Poisson process through its counting and inter-arrival properties. Students derive arrival time distributions and apply the process to real systems.

Lesson 4 • Reliability and Survival Analysis

Applies exponential and Weibull distributions to model component lifetimes. Students compute failure rates, survival functions, and system reliability.

Lesson 5 • Discrete-Time Markov Chains

Introduces the Markov property and transition probability matrices. Students compute multi-step transition probabilities and classify states.