From classical foundations to advanced modern theory, this self-contained and comprehensive guide to probability weaves together mathematical proofs, historical context and richly detailed illustrative applications. A theorem discovery approach is used throughout, setting each proof within its historical setting and is accompanied by a consistent emphasis on elementary methods of proof. Each topic is presented in a modular framework, combining fundamental concepts with worked examples, problems and digressions which, although mathematically rigorous, require no specialised or advanced mathematical background. Augmenting this core material are over 80 richly embellished practical applications of probability theory, drawn from a broad spectrum of areas both classical and modern, each tailor-made to illustrate the magnificent scope of the formal results. Providing a solid grounding in practical probability, without sacrificing mathematical rigour or historical richness, this insightful book is a fascinating reference and essential resource, for all engineers, computer scientists and mathematicians.

for excessive generality when embarking upon the process of constructing a formal theory and developing an appreciation of its applications. I shall accordingly restrict myself in this book (for the most part) to providing an honest account of the theory and applications of chance experiments in discrete and continuous spaces. 4 Sets and operations on sets An abstract sample space Ω is an aggregate (or set) of sample points ω. We identify events of interest with subsets of the space at hand.

is purely for later algebraic convenience.) In random mating, 78 III.3 An application in genetics, Hardy’s law (AA,AA) (AA,Aa) AA (AA,aa) (Aa,AA) (Aa,Aa) Aa (Aa,aa) (aa,AA) (aa,Aa) aa (aa,aa) Figure 3: Genotype combinations. which is the only case we consider here, each offspring in the ﬁlial generation is assumed to arise out of a mating of two randomly selected parents from the population (the sampling is with replacement). During mating each parent contributes a randomly selected

the binomial which, in view of its historical signiﬁcance as well as its practical importance, is worth enshrining. T HEOREM 3 Suppose λ > 0 is a ﬁxed constant and { πn , n ≥ 1 } is a sequence of positive numbers, 0 < πn < 1, satisfying nπn → λ. Then bn (m; πn ) → p(m; λ) for each ﬁxed positive integer m. A limit theorem such as this applies formally not to a ﬁxed collection of events but, rather, to a whole sequence of event collections indexed by a parameter n. Its practical sway is seen in

or appropriate. for ces, bes, bessie, mummyyummyyum, mozer, muti, cecily stewart venkatesh, anil, boneil, banana peel, studebaker, suffering succotash, anna, banana, baby dinosaur, bumpus, pompous, boom boom music, twelve ways of biscuit, booboo, binjammin, rumpus, gumpus, grumpus, goofy, goofus, anil benjamin venkatesh, chachu, achu, chach, ach, uch, uchie, sweetie, honey, bachu, joy, achoo tree, muggle artefacts, anna did it, bump, lump, sump, chachubachubach, stormcloud, jammie, sara, baby

Explain how the bent binary digits zk (t; p) (k ≥ 1) can be used to construct a model for independent tosses of an unfair coin whose success probability is p and failure probability is q = 1 − p. 16. Bernstein polynomials. Suppose f(t) is a continuous function on the unit interval. Show that 1 f 0 “ z (t; x) + · · · + z (t; x) ” n 1 dt = n n f k=0 “k” xk (1 − x)n−k . n For each n, the expression Bn (x) given by the sum on the right deﬁnes a function on the unit interval 0 ≤ x ≤ 1 [with the