Course notes and worked problems for mathematics and statistics students.
For the sake of brevity, we will always represent this number 2.718281828459… by the letter e. Leonhard Euler
The first-year groundwork: sets and the algebra of sets, sets of numbers, complex numbers and surds, relations and functions, quadratics, equations and inequalities, trigonometry, limits and differentiation, polynomials, exponentials and logarithms, curve sketching, induction and the binomial expansion, integration, matrices and vectors.
Sets and numbers, functions, polynomial and rational functions, trigonometry, exponentials and logarithms, differentiation and integration, then descriptive statistics and probability. Every tutorial-sheet question is worked in full.
Probability axioms, conditional probability and Bayes' theorem, counting techniques, random variables, and the standard discrete and continuous distributions.
The machinery behind the limit theorems. Generating functions turn the convolution of independent sums into a product, and uniqueness turns the product back into a distribution. Conditioning gives the compound distributions and the martingales. Four distinct senses of convergence are separated, each converse settled by a counterexample, and the central limit theorem is then proved in the one that is meant. The course closes with stochastic processes, Markov chains and simulation. Every result is proved and every problem worked in full.
Randomness indexed by time. Conditioning comes first, because everything after it is an expectation computed one step at a time, and the Chapman–Kolmogorov equation is what that buys. States are then classified as transient or recurrent, and the random walk answers the question the classification was built for: it returns in one and two dimensions and escapes in three. Every result is proved and every problem worked in full.
How a response depends on one or more regressors, from the straight line to the full linear model in matrix form. Least squares and its properties, tests and intervals, then checking the model: residuals, lack of fit and influential points. Transformations and weighting, choosing among models, multicollinearity and ridge regression, dummy variables and polynomials, and on to logistic and Poisson regression, non-linear models, autocorrelated errors and calibration. Every formula is proved and every method is worked on numbers.
Data where the order of the observations is the whole point. Stationarity and the autocorrelation function first, then the AR, MA and ARMA models built on them, and differencing to reach the ones that are not stationary. The partial autocorrelation makes identification possible; estimation, diagnostic checking and forecasting complete the Box–Jenkins cycle, and seasonal models extend it; the spectrum shows the same series as a mixture of cycles; state space models and the Kalman filter handle what ARMA cannot; and ARCH addresses the variance that every earlier model assumed constant. Every problem is worked in full.
Describing data, sampling distributions, estimation and confidence intervals, hypothesis testing and P-values, analysis of variance, and linear regression and correlation.
Geometry done by algebra, then the calculus built on it. The conic sections are derived from their focal definitions and reunified by eccentricity, collapsing into a single polar equation. Limits lead to the mean value theorems, and through them to series and curvature; integration is developed as technique and then applied to area, volume, arc length and surface. Vectors carry the subject into three dimensions and into motion, and the course closes with differential equations. Every result is proved and the tutorial questions are worked in full.
Calculus carried into several variables and then into three dimensions. Partial derivatives give the gradient and the tangent plane, and the Jacobian that measures how a change of variables stretches area returns as the factor in the multiple integral. Vector fields bring the divergence and the curl, and the course converges on Green’s, Stokes’ and the divergence theorems — one theorem in three dresses, each saying that integrating a derivative over a region equals integrating the original over its boundary. It closes with Laplace transforms and Fourier series. All 114 tutorial questions are worked in full.
Matrices and row reduction built up until they become statements about linear maps rather than procedures. Determinants are developed as a test for invertibility; rank–nullity ties what a map collapses to what it reaches; and the course ends with the spectral theorem, which makes every real symmetric matrix diagonalizable and every quadratic form a sum of squares. Every result is proved, and all 64 practice problems are fully worked.
Structure studied for its own sake. Groups first, where Lagrange’s theorem is the earliest result that could not have been guessed from the definitions, then the quotient constructions that make homomorphisms intelligible. Rings follow, and the discovery that ideals rather than subrings are what quotients need. It closes with fields and the polynomial quotients that build them. All 62 results are proved and every problem worked in full.
Distributions of functions of random variables and order statistics, estimation by moments, least squares and maximum likelihood, properties of estimators, and the theory of hypothesis testing.
What makes one estimator better than another, and how far the question can be pushed. Sufficiency reduces the data without losing information; completeness makes the reduction unique, and the two together deliver the minimum-variance unbiased estimator. Maximum likelihood is then developed with its score, its information and the bound that information places on any unbiased estimator. The last chapter turns to testing, from the Neyman–Pearson lemma to the three large-sample tests that agree asymptotically and differ in what they ask you to compute. Every result is proved and all 56 problems are worked in full.
The complex plane and power series, analytic functions and the Cauchy–Riemann equations, conformal mappings, contour integration and the Cauchy–Goursat theorem, the maximum modulus principle, the calculus of residues, and analytic continuation through to the monodromy theorem.
An undergraduate first course in the subject. Metric spaces are developed as far as completeness, then put to work: the contraction mapping theorem yields Picard’s existence theorem for differential equations, and the Baire category theorem follows. Norms and inner products are then added to the vector space structure, giving Banach and Hilbert spaces, the Hahn–Banach theorem, the projection theorem and the Riesz representation theorem. Every result is proved and every practice problem is fully worked.
What survives when distance is thrown away. Metric spaces come first, then the observation that every proof about continuity used only the open sets — so the open sets alone are taken as the definition. Compactness and connectedness follow, and with them the theorems of a first analysis course reappear as statements about spaces: the intermediate value theorem becomes a remark about connectedness, and the extreme value theorem one about compactness. Every result is proved and every problem worked in full.
The calculus rebuilt on the completeness of the real line. Open and closed sets come first, then compactness and connectedness, and from those two the theorems everyone already knows — intermediate value, extreme value, mean value — fall out as consequences rather than assertions. Bounded variation and the Riemann–Stieltjes integral close it. Every one of the 98 results is proved and every problem worked in full.
How to compute with real numbers on a machine, and how to know how wrong the answer can be. The course starts from floating-point arithmetic and error, then covers root finding (bisection, fixed-point iteration, Newton and secant methods), complex roots of polynomials, direct and iterative methods for linear systems, interpolation, splines and least squares, numerical differentiation and integration, and Euler and Runge–Kutta methods for differential equations. Every method comes with a convergence or error theorem and its proof, and every example is worked to the last digit.
Distribution-free inference: the sign test and confidence intervals for the median, Wilcoxon signed-rank and rank-sum, Kruskal–Wallis, Kolmogorov–Smirnov, rank correlation by Spearman and Kendall, tests of randomness, and non-parametric trend detection by Mann–Kendall and Theil–Sen.
Matrix algebra and the geometry of random vectors, the multivariate normal distribution with Hotelling’s T2 and Wilks’ lambda, classification by Fisher’s discriminant, multivariate analysis of variance, and multivariate multiple regression.
The analysis of variance and regression treated as one subject. Quadratic forms and Cochran’s theorem supply the distribution theory the F test rests on; estimability says which parameters a rank-deficient design can speak about; and the general linear hypothesis reduces every test in the course to one. Ends with regression diagnostics and variable selection.
The statistics of time to an event when many times are censored. The course builds the survival, hazard and cumulative hazard functions and parametric models fitted with censored data, then the Kaplan–Meier and Nelson–Aalen estimators, life tables and the log-rank test. It develops the Cox proportional hazards model from its partial likelihood, fits and checks it with Schoenfeld residuals, and goes on to accelerated failure time models, competing risks, frailty, recurrent events and the sample size of survival trials, ending with a complete case study of the Mayo PBC trial. Every example uses real data sets, the 38 software panels run in R with the survival package, and all 75 problems are worked in full.
The theory beneath the methods. Order statistics and their exact distributions, statistics that are distribution-free over a class, and Hoeffding’s theory of U-statistics with the projection principle that gives them their asymptotic normality.
How to collect data so that the analysis can answer the question. Built on the linear model, the course runs from the one-way analysis of variance and its contrasts to block designs, Latin squares and balanced incomplete blocks. It then covers factorial experiments: the 2k design, confounding and fractions of given resolution. The last part treats random and mixed effects, nested and split-plot designs, and the choice of the right error term. Every expected mean square is proved, and all 74 problems are worked in full.
How to plan, analyse and interpret studies of disease in populations. The course starts from the designs and what each can estimate, then covers measuring disease and exposure, the measures of frequency and association with their intervals, and attributable risk. It goes on to bias, confounding and causal diagrams, standardisation, stratified analysis and the Mantel–Haenszel procedure, life tables and person-years, and logistic and Poisson regression. It ends by comparing every method of adjustment on the same data. Every estimator is derived, and all 81 problems are worked in full.
The linear model and everything that goes wrong with it. The course builds least squares, its large-sample theory and inference in matrix form, then relaxes each assumption in turn: specification error and structural breaks, maximum likelihood and GLS, heteroscedasticity, autocorrelation and ARCH, unit roots and cointegration, instrumental variables, simultaneous systems and VARs, GMM, panel data, and limited dependent variables. Every result is proved, and the 61 software panels run real data in R, with the Stata and EViews commands alongside. All 117 problems are worked in full.
From explicit solution methods to the qualitative theory. The course proves existence and uniqueness, continuous dependence and blow-up, then develops linear systems, the matrix exponential, Laplace transforms and Floquet theory, series solutions and special functions, and Sturm–Liouville theory. It goes on to phase planes, Lyapunov functions, limit cycles, bifurcations and chaos, and ends with perturbation and numerical methods. Every theorem is proved, the 42 software panels run in Python with SciPy and SymPy, and all 87 problems are worked in full.
The wave, heat and Laplace equations and the ideas that unite them. The course starts from characteristics and shock waves, classifies second-order equations and asks which problems are well posed, then solves them by d'Alembert, Kirchhoff and Poisson formulas, heat kernels, Fourier series, Green's functions, transforms and separation in round regions. Maximum principles and energy give uniqueness and stability, the Dirichlet principle gives weak solutions and finite elements, and the course ends with finite differences, travelling fronts and Turing patterns. Every theorem is proved, the 42 software panels run in Python with NumPy, SciPy and SymPy, and all 82 problems are worked in full.
Optimisation under constraints, from linear programming outwards. The course builds LP models, proves the geometry and the simplex method, develops duality and sensitivity analysis, and solves transportation, assignment and network problems through total unimodularity. It goes on to integer, non-linear and dynamic programming, then to decisions under uncertainty: decision trees, games, inventories, queues, Markov decision processes and simulation. Worked examples include problems from Sharma; every theorem is proved, the 48 software panels run in Python with SciPy and HiGHS, and all 95 problems are worked in full.
Improper integrals and the special functions they define — gamma, beta, error and the incomplete forms that carry the classical statistical distributions. Then the orthogonal polynomials and the Sturm–Liouville theory that shows why all of them are orthogonal for one reason, Fourier series and the transform they become on an unbounded domain, and the Laplace transform.
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