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.
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; 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.
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 $T^2$ 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 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.
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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