Probability

Product and sum rules, permutations with and without repetition, combinations, and multinomial coefficients.

Counting unions of overlapping sets, derangements, and the pigeonhole principle as a proof device.

Assumes: Counting, Permutations and Combinations

Experiments, outcomes, events and sigma-algebras; Kolmogorov's three axioms and what follows from them.

Assumes: Counting, Permutations and Combinations

The difference between independent and mutually exclusive events — the single most common exam trap.

Assumes: Sample Spaces and the Axioms of Probability

Reading probability tables, the multiplication rule, and marginalising a joint distribution.

Assumes: Events, Independence and Mutual Exclusivity

Partitioning a sample space to compute awkward probabilities, with tree diagrams.

Assumes: Joint, Marginal and Conditional Probability

Inverting conditional probabilities, prior/likelihood/posterior, base-rate fallacies and medical-test paradoxes.

Assumes: The Law of Total Probability

Mapping outcomes to numbers, discrete vs continuous, and the support of a random variable.

Assumes: Sample Spaces and the Axioms of Probability

The three ways to describe a distribution, how to convert between them, and why densities are not probabilities.

Assumes: Random Variables

Expected value for discrete and continuous variables, linearity, and the law of the unconscious statistician.

Assumes: PMF, PDF and CDF

E[X|Y] as a random variable, the tower property, and conditional variance decomposition.

Assumes: Expectation

Variance and standard deviation, higher moments, and using MGFs to derive distributions of sums.

Assumes: Expectation

Measuring joint variation, Pearson correlation, and why zero correlation does not imply independence.

Assumes: Variance, Moments and Generating Functions

Single trials and counts of successes: PMF derivation, moments, and the normal approximation.

Assumes: Variance, Moments and Generating Functions

Waiting times, counts until r successes, and sampling without replacement.

Assumes: Bernoulli and Binomial Distributions

Rare events, derivation as a binomial limit, the Poisson process, and additivity.

Assumes: Bernoulli and Binomial Distributions

Continuous uniform sampling, the exponential distribution, and its memoryless property.

Assumes: PMF, PDF and CDF

Sums of exponentials, the gamma function, and the beta distribution as a conjugate prior over probabilities.

Assumes: Uniform and Exponential Distributions

The Gaussian density, standardisation, z-tables, and the 68–95–99.7 rule with worked lookups.

Assumes: Variance, Moments and Generating Functions

The three sampling distributions behind inference: definitions, degrees of freedom and interrelations.

Assumes: The Normal Distribution

The CDF method, change-of-variables with Jacobians, and inverse transform sampling.

Assumes: PMF, PDF and CDF

Joint densities, conditional density functions, marginalisation by integration and independence.

Assumes: Joint, Marginal and Conditional Probability

Distribution of a sum via convolution and MGFs, and the closure properties of common families.

Assumes: Variance, Moments and Generating Functions

Mean vectors and covariance matrices, contours of constant density, marginals and conditionals.

Assumes: The Normal Distribution · Quadratic Forms and Definiteness

Markov, Chebyshev, Jensen, Cauchy–Schwarz and Hoeffding bounds, and where each is used in ML theory.

Assumes: Variance, Moments and Generating Functions

Weak and strong laws, convergence in probability vs almost surely, and what they justify about sample means.

Assumes: Probability Inequalities

Why sample means go normal regardless of the parent distribution, with simulation evidence and sample-size rules.

Assumes: Laws of Large Numbers

Transition matrices, Chapman–Kolmogorov, classification of states, and stationary distributions.

Assumes: Joint, Marginal and Conditional Probability · Eigenvalues and Eigenvectors