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"There is a logical flaw in the statistical methods used across experimental science. This fault is not just a minor academic quibble: it underlies a reproducibility crisis now threatening entire disciplines. In an increasingly data-reliant culture, this same deeply rooted error shapes decisions in medicine, law, and public policy with profound consequences. The foundation of the problem is a misunderstanding of probability and our ability to make inferences from data. Aubrey Clayton traces the history of how statistics went astray, beginning with the groundbreaking work of the seventeenth-century mathematician Jacob Bernoulli and winding through gambling, astronomy, and genetics. He recounts the feuds among rival schools of statistics, exploring the surprisingly human problems that gave rise to the discipline and the all-too-human shortcomings that derailed it. Clayton highlights how influential nineteenth- and twentieth-century figures developed a statistical methodology they claimed was purely objective in order to silence critics of their political agendas, including eugenics. Clayton provides a clear account of the mathematics and logic of probability, conveying complex concepts accessibly for readers interested in the statistical methods that frame our understanding of the world. He contends that we need to take a Bayesian approach-incorporating prior knowledge when reasoning with incomplete information-in order to resolve the crisis. Ranging across math, philosophy, and culture, Bernoulli's Fallacy explains why something has gone wrong with how we use data-and how to fix it"--

Probabilities --- Mathematical statistics --- Binomial distribution. --- Law of large numbers. --- MATHEMATICS / History & Philosophy --- Influence (Literary, artistic, etc.) --- Philosophy --- Philosophy. --- Bernoulli, Jakob, --- Influence. --- 1800-1999 --- Bayesian statistics. --- frequentist statistics. --- history of math. --- history of statistics. --- probability. --- replication crisis. --- statistics and science. --- statistics. --- Large numbers, Law of --- Numbers, Large --- Convergence --- Artistic impact --- Artistic influence --- Impact (Literary, artistic, etc.) --- Literary impact --- Literary influence --- Literary tradition --- Tradition (Literature) --- Art --- Influence (Psychology) --- Literature --- Intermediality --- Intertextuality --- Originality in literature --- Bernoulli distribution --- Distribution, Bernoulli --- Distribution, Binomial --- Binomial theorem --- Distribution (Probability theory) --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Sampling (Statistics) --- Probability --- Combinations --- Chance --- Least squares --- Risk --- Statistical methods --- Bernoulli, Jacques, --- Bernoullius, Jacobus, --- Bernulli, I︠A︡kob, --- Bernoulli, Jacob,

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In this classic of statistical mathematical theory, Harald Cramér joins the two major lines of development in the field: while British and American statisticians were developing the science of statistical inference, French and Russian probabilitists transformed the classical calculus of probability into a rigorous and pure mathematical theory. The result of Cramér's work is a masterly exposition of the mathematical methods of modern statistics that set the standard that others have since sought to follow. For anyone with a working knowledge of undergraduate mathematics the book is self contained. The first part is an introduction to the fundamental concept of a distribution and of integration with respect to a distribution. The second part contains the general theory of random variables and probability distributions while the third is devoted to the theory of sampling, statistical estimation, and tests of significance.

Mathematical statistics --- 519.2 --- 519.2 Probability. Mathematical statistics --- Probability. Mathematical statistics --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Statistical methods --- Statistique mathématique --- Mathematical statistics. --- Statistique mathématique --- Statistique mathématique. --- Distribution (théorie des probabilités) --- Distribution (Probability theory) --- A priori probability. --- Addition theorem. --- Additive function. --- Analysis of covariance. --- Arithmetic mean. --- Axiom. --- Bayes' theorem. --- Bias of an estimator. --- Binomial distribution. --- Binomial theorem. --- Bolzano–Weierstrass theorem. --- Borel set. --- Bounded set (topological vector space). --- Calculation. --- Cartesian product. --- Central moment. --- Characteristic function (probability theory). --- Characteristic polynomial. --- Coefficient. --- Commutative property. --- Confidence interval. --- Convergence of random variables. --- Correlation coefficient. --- Degeneracy (mathematics). --- Degrees of freedom (statistics). --- Diagram (category theory). --- Dimension. --- Distribution (mathematics). --- Distribution function. --- Empirical distribution function. --- Equation. --- Estimation theory. --- Estimation. --- Identity matrix. --- Independence (probability theory). --- Interval (mathematics). --- Inverse probability. --- Invertible matrix. --- Joint probability distribution. --- Laplace distribution. --- Lebesgue integration. --- Lebesgue measure. --- Lebesgue–Stieltjes integration. --- Likelihood function. --- Limit (mathematics). --- Linear regression. --- Logarithm. --- Logarithmic derivative. --- Logarithmic scale. --- Marginal distribution. --- Mathematical analysis. --- Mathematical induction. --- Mathematical theory. --- Mathematics. --- Matrix (mathematics). --- Maxima and minima. --- Measure (mathematics). --- Method of moments (statistics). --- Metric space. --- Minor (linear algebra). --- Moment (mathematics). --- Moment matrix. --- Normal distribution. --- Numerical analysis. --- Parameter. --- Parity (mathematics). --- Poisson distribution. --- Probability distribution. --- Probability theory. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random variable. --- Realization (probability). --- Riemann integral. --- Sample space. --- Sampling (statistics). --- Scientific notation. --- Series (mathematics). --- Set (mathematics). --- Set function. --- Sign (mathematics). --- Standard deviation. --- Statistic. --- Statistical Science. --- Statistical hypothesis testing. --- Statistical inference. --- Statistical regularity. --- Statistical theory. --- Subset. --- Summation. --- Theorem. --- Theory. --- Transfinite number. --- Uniform distribution (discrete). --- Variable (mathematics). --- Variance. --- Weighted arithmetic mean. --- Z-test. --- Distribution (théorie des probabilités)

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We use addition on a daily basis-yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research.Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series-long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+. . .=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms-the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat's Last Theorem.Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.

Number theory. --- Mathematics --- Number study --- Numbers, Theory of --- Algebra --- Absolute value. --- Addition. --- Analytic continuation. --- Analytic function. --- Automorphic form. --- Axiom. --- Bernoulli number. --- Big O notation. --- Binomial coefficient. --- Binomial theorem. --- Book. --- Calculation. --- Chain rule. --- Coefficient. --- Complex analysis. --- Complex number. --- Complex plane. --- Computation. --- Congruence subgroup. --- Conjecture. --- Constant function. --- Constant term. --- Convergent series. --- Coprime integers. --- Counting. --- Cusp form. --- Determinant. --- Diagram (category theory). --- Dirichlet series. --- Division by zero. --- Divisor. --- Elementary proof. --- Elliptic curve. --- Equation. --- Euclidean geometry. --- Existential quantification. --- Exponential function. --- Factorization. --- Fourier series. --- Function composition. --- Fundamental domain. --- Gaussian integer. --- Generating function. --- Geometric series. --- Geometry. --- Group theory. --- Hecke operator. --- Hexagonal number. --- Hyperbolic geometry. --- Integer factorization. --- Integer. --- Line segment. --- Linear combination. --- Logarithm. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Matrix group. --- Modular form. --- Modular group. --- Natural number. --- Non-Euclidean geometry. --- Parity (mathematics). --- Pentagonal number. --- Periodic function. --- Polynomial. --- Power series. --- Prime factor. --- Prime number theorem. --- Prime number. --- Pythagorean theorem. --- Quadratic residue. --- Quantity. --- Radius of convergence. --- Rational number. --- Real number. --- Remainder. --- Riemann surface. --- Root of unity. --- Scientific notation. --- Semicircle. --- Series (mathematics). --- Sign (mathematics). --- Square number. --- Square root. --- Subgroup. --- Subset. --- Sum of squares. --- Summation. --- Taylor series. --- Theorem. --- Theory. --- Transfinite number. --- Triangular number. --- Two-dimensional space. --- Unique factorization domain. --- Upper half-plane. --- Variable (mathematics). --- Vector space.

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The legendary Renaissance math duel that ushered in the modern age of algebraThe Secret Formula tells the story of two Renaissance mathematicians whose jealousies, intrigues, and contentious debates led to the discovery of a formula for the solution of the cubic equation. Niccolò Tartaglia was a talented and ambitious teacher who possessed a secret formula—the key to unlocking a seemingly unsolvable, two-thousand-year-old mathematical problem. He wrote it down in the form of a poem to prevent other mathematicians from stealing it. Gerolamo Cardano was a physician, gifted scholar, and notorious gambler who would not hesitate to use flattery and even trickery to learn Tartaglia's secret.Set against the backdrop of sixteenth-century Italy, The Secret Formula provides new and compelling insights into the peculiarities of Renaissance mathematics while bringing a turbulent and culturally vibrant age to life. It was an era when mathematicians challenged each other in intellectual duels held outdoors before enthusiastic crowds. Success not only enhanced the winner's reputation, but could result in prize money and professional acclaim. After hearing of Tartaglia's spectacular victory in one such contest in Venice, Cardano invited him to Milan, determined to obtain his secret by whatever means necessary. Cardano's intrigues paid off. In 1545, he was the first to publish a general solution of the cubic equation. Tartaglia, eager to take his revenge by establishing his superiority as the most brilliant mathematician of the age, challenged Cardano to the ultimate mathematical duel.A lively and compelling account of genius, betrayal, and all-too-human failings, The Secret Formula reveals the epic rivalry behind one of the fundamental ideas of modern algebra.

Equations --- Algebra --- Mathematics --- History. --- Ars Magna. --- Arturo Sangalli. --- Blaise Pascal. --- Cardano-Tartaglia formula. --- Carl Friedrich Gauss. --- Euler's Pioneering Equation. --- Ian Stewart. --- Isaac Newton. --- Italian Renaissance. --- Joseph-Louis Lagrange. --- Leibniz. --- Lodovico Ferrari. --- Maria Gaetana Agnesi. --- Omar Khayyám. --- Pierre de Fermat. --- Rafael Bombelli. --- Renaissance men. --- Robin Wilson. --- Scipione del Ferro. --- Significant Figures. --- algebra. --- binomial coefficients. --- binomial theorem. --- coefficients. --- imaginary numbers. --- negative numbers. --- roots of the cubic function. --- Tartaglia, Niccolò, --- Cardano, Girolamo, --- Cardano, Girolamo --- Tartaglia, Niccolò --- 1500-1599 --- Tartaglia, Nicolaus --- Fontana, Niccolò --- Tartalea, Niccolò --- Tartaglia, Nicolas --- Tartalea, Nicolo --- Tartaglia, Nicolo --- Tartalea --- Mathematiker --- Rechenmeister --- Übersetzer --- Ingenieur --- Lehrer --- Prof. --- Brescia --- Venedig --- Verona --- 1500-1557 --- Cardano, Geronimo --- Cardano, Hieronimo --- Cardano, Hieronymus --- Cardano --- Cardani, Hieronymus --- Cardanus, Girolamo --- Cardanus, Hieronimus Castellioneus --- Cardanus, Hieronymus Castellioneus --- Cardanus, Hieronimus C. --- Cardanus, Hier. --- Cardanus --- Cardanus, H. --- Cardini, Hieronymus --- Giralmo --- Hieronymus --- Kardano, Dzhirolamo --- Cardano, Gerolamo --- Cardano, Giralmo --- Cardan, Jérôme --- Cardan, Hierosme --- Cardanus, Hieronymus C. --- Cardanus, Hieronymus --- Cardanus, Hierosme --- Castellioneus, Hieronymus --- Geronimo --- Girolamo --- Jérome --- Cardan, Girolamo --- Cardano, Hieronimus --- Jérôme --- Cardan --- Cardan, H. --- Naturwissenschaftler --- Astronom --- Philosoph --- Arzt --- Pavia --- Rom --- Padua --- Mailand --- Bologna --- Arnaldus --- Benvenutus Rambaldi --- Caesius, Georg --- Cardano, Giovanni Battista --- Cigalini, Francesco --- Dasypodius, Konrad --- Frölich, Huldrich --- Majoragio, Marcantonio --- Petrarca, Francesco --- Ursinus, Adam --- 24.09.1501-20.09.1576 --- 24.11.1501-01.09.1575 --- 1501-1576 --- 1501-1575 --- Fontana, Nicolò, --- Tartaglia, Nicholas, --- Tartaglia, Nicolas, --- Tartaglia, Nicolò, --- Tartaia, Nicolò, --- Tartalea, Nicolai, --- Cardan, H., --- Cardan, Jérôme, --- Cardano, Gerolamo, --- Cardano, Hieronimo, --- Cardanus, Hierome, --- Cardanus, Hieronymus, --- Cardanus, Hierosme, --- Kardano, Dzhirolamo, --- Cardan, Jerome --- Cardan, Jerome,