An Introduction to Benford's Law
Theodore P. Hill, Arno Berger
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law’s colorful history, rapidly growing body of empirical evidence, and wide range of applications.
An Introduction to Benford’s Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford’s law.
Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.
Approximation error, Distribution function, Theorem, Subset, Natural number, Probability measure, Probability space, Initial value problem, Law of large numbers, Mathematics, Proportionality (mathematics), Eigenvalues and eigenvectors, Number theory, Random variable, Result, Linear map, Continuous function, Countable set, Empty set, Calculation, Round-off error, Empirical distribution function, Sampling (statistics), Theory, Coefficient, Approximation, Monotonic function, Dimension, Power set, Real number, Positive real numbers, Equation, Division by zero, Linear difference equation, Independent and identically distributed random variables, Almost surely, Open problem, Parameter, Sign (mathematics), Summation, Borel set, Existential quantification, Integer, Explanation, Convergence of random variables, Differential equation, Newton's method, Statistic, Rate of convergence, Absolute continuity, Fibonacci number, Algorithm, Probability theory, Nonnegative matrix, Discrete time and continuous time, Ergodic theory, Normal distribution, Benford's law, Uniform distribution (discrete), Data set, Significant figures, Logarithmic distribution, Natural density, Probability, Logarithm, Probability distribution, Special case, Prime number, Lebesgue measure, Significand