About This Course

This book teaches the mathematics of uncertainty when the random mechanism is known. A die is fair, a test has stated error rates, a waiting time has a given density: from such a model we deduce how likely events are, what values to expect, how quantities depend on one another, and what happens when many random contributions are added. It is written for a reader working alone, typically a second-year undergraduate in mathematics, statistics, computer science, engineering, economics, or a natural science, who wants to be able to calculate and also to say why each calculation is legitimate.

Three things are assumed. From Calculus: derivatives, definite and improper integrals, substitution, integration by parts, geometric and exponential series, and Taylor expansion to second order. From Discrete Mathematics: sets, functions, induction, and the idea of counting by a product of choices. From Multivariable Calculus, needed only from Chapter 5 onward: double integrals over simple regions and the change-of-variables formula in the plane. When a theorem from another course is used, it is stated with the hypotheses the argument needs and marked once with its library identifier; it is not proved again here.

The book is proof-aware without being measure-theoretic. Every result that can be proved with sums, integrals, and careful bookkeeping is proved. A handful of results need tools this course does not own: the uniqueness theorem for generating functions, the continuity theorem that converts convergence of transforms into convergence of distributions, and the strong law of large numbers in full generality. Those are stated exactly, their role is explained, and the reader is told where the proof lives, usually Measure-Theoretic Probability.

Six abilities mark the end of the book, each to be exercised with the page closed. Build a probability model for a verbal problem and say which assumptions you have made. Condition correctly, in both directions, and decide whether events or random variables are independent. Compute expectations, variances, and covariances, using linearity and indicators where they shorten the work. Find the distribution of a transformed or summed random quantity by the method that suits it: distribution functions, densities, convolution, or a generating function. Bound a probability with an inequality and say how sharp the bound is. Use the laws of large numbers and the central limit theorem with their hypotheses in view, and label every numerical conclusion as exact, a guaranteed bound, or an approximation.

Three neighbouring subjects are deliberately left alone. Drawing conclusions about an unknown model from observed data, including estimation, confidence statements, and tests, belongs to Mathematical Statistics. Random systems that evolve in time, such as Markov chains and Poisson processes, belong to Stochastic Processes. The construction of probability from measure and the proofs that depend on it belong to Measure-Theoretic Probability. Where this book touches one of those borders, it says so.

Notation is conventional. The probability of an event \(A\) is \(P(A)\) ; its complement is \(A^c\) . Random variables are capital letters, their values lower-case letters. Expectation, variance, and covariance are \(E[X]\) , \(\operatorname{Var}(X)\) , and \(\operatorname{Cov}(X,Y)\) . The indicator of an event \(A\) is \(\mathbf 1_A\) . A distribution function is \(F\) , a probability mass function is \(p\) , a density is \(f\) , each with the variable as a subscript when more than one is in play. The standard normal distribution function is \(\Phi\) . Logarithms are natural.

Each chapter ends with exercises that run from recall to proof. Work them with pencil before looking anything up. The companion solutions give full arguments for some, graduated hints for others, and bare answers for a few; an exercise with no entry is meant to be settled by the methods of its chapter alone. Several exercises present a plausible argument and ask you to find the flaw. Probability is unusually rich in convincing wrong arguments, and learning to locate the error is as much a part of the subject as learning the right method. The cumulative review at the end mixes the chapters without saying which one each problem belongs to, because choosing the tool is part of the problem.