1.
Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
2.
Normal distribution
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In probability theory, the normal distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. The normal distribution is useful because of the limit theorem. Physical quantities that are expected to be the sum of independent processes often have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are normally distributed, the normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped, the probability density of the normal distribution is, f =12 π σ2 e −22 σ2 Where, μ is mean or expectation of the distribution. σ is standard deviation σ2 is variance A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate. The simplest case of a distribution is known as the standard normal distribution. The factor 1 /2 in the exponent ensures that the distribution has unit variance and this function is symmetric around x =0, where it attains its maximum value 1 /2 π and has inflection points at x = +1 and x = −1. Authors may differ also on which normal distribution should be called the standard one, the probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a normal deviate, then X = Zσ + μ will have a normal distribution with expected value μ. Conversely, if X is a normal deviate, then Z = /σ will have a standard normal distribution. Every normal distribution is the exponential of a function, f = e a x 2 + b x + c where a is negative. In this form, the mean value μ is −b/, for the standard normal distribution, a is −1/2, b is zero, and c is − ln /2. The standard Gaussian distribution is denoted with the Greek letter ϕ. The alternative form of the Greek phi letter, φ, is used quite often. The normal distribution is often denoted by N. Thus when a random variable X is distributed normally with mean μ and variance σ2, some authors advocate using the precision τ as the parameter defining the width of the distribution, instead of the deviation σ or the variance σ2
3.
Hypergeometric distribution
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In contrast, the binomial distribution describes the probability of k successes in n draws with replacement. In statistics, the hypergeometric test uses the hypergeometric distribution to calculate the significance of having drawn a specific k successes from the aforementioned population. The test is used to identify which sub-populations are over- or under-represented in a sample. This test has a range of applications. For example, a group could use the test to understand their customer base by testing a set of known customers for over-representation of various demographic subgroups. The following conditions characterize the distribution, The result of each draw can be classified into one of two mutually exclusive categories. The probability of a success changes on each draw, as each draw decreases the population, the pmf is positive when max ≤ k ≤ min. The pmf satisfies the recurrence relation P = P with P =, as one would expect, the probabilities sum up to 1, ∑0 ≤ k ≤ n =1 This is essentially Vandermondes identity from combinatorics. Also note the following identity holds, = and this follows from the symmetry of the problem, but it can also be shown by expressing the binomial coefficients in terms of factorials and rearranging the latter. The classical application of the distribution is sampling without replacement. Think of an urn with two types of marbles, red ones and green ones, define drawing a green marble as a success and drawing a red marble as a failure. If the variable N describes the number of all marbles in the urn and K describes the number of green marbles, in this example, X is the random variable whose outcome is k, the number of green marbles actually drawn in the experiment. This situation is illustrated by the following table, Now. Standing next to the urn, you close your eyes and draw 10 marbles without replacement, what is the probability that exactly 4 of the 10 are green. This problem is summarized by the following table, The probability of drawing exactly k green marbles can be calculated by the formula P = f =. Hence, in this example calculate P = f = =5 ⋅814506010272278170 =0.003964583 …, intuitively we would expect it to be even more unlikely for all 5 marbles to be green. P = f = =1 ⋅122175910272278170 =0.0001189375 …, As expected, in Holdem Poker players make the best hand they can combining the two cards in their hand with the 5 cards eventually turned up on the table. The deck has 52 and there are 13 of each suit, for this example assume a player has 2 clubs in the hand and there are 3 cards showing on the table,2 of which are also clubs
4.
Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω
5.
John Wiley & Sons
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Founded in 1807, Wiley is also known for publishing For Dummies. As of 2015, the company had 4,900 employees, Wiley was established in 1807 when Charles Wiley opened a print shop in Manhattan. Wiley later shifted its focus to scientific, technical, and engineering subject areas, Charles Wileys son John took over the business when his father died in 1826. The firm was successively named Wiley, Lane & Co. then Wiley & Putnam, the company acquired its present name in 1876, when Johns second son William H. Wiley joined his brother Charles in the business. Through the 20th century, the company expanded its activities, the sciences. Since the establishment of the Nobel Prize in 1901, Wiley and its companies have published the works of more than 450 Nobel Laureates. Wiley in December 2010 opened an office in Dubai, to build on its business in the Middle East more effectively, the company has had an office in Beijing, China, since 2001, and China is now its sixth-largest market for STEM content. Wiley established publishing operations in India in 2006, and has established a presence in North Africa through sales contracts with academic institutions in Tunisia, Libya, and Egypt. On April 16,2012, the announced the establishment of Wiley Brasil Editora LTDA in São Paulo, Brazil. Wileys scientific, technical, and medical business was expanded by the acquisition of Blackwell Publishing in February 2007. Through a backfile initiative completed in 2007,8.2 million pages of content have been made available online. Other major journals published include Angewandte Chemie, Advanced Materials, Hepatology, International Finance, launched commercially in 1999, Wiley InterScience provided online access to Wiley journals, major reference works, and books, including backfile content. Journals previously from Blackwell Publishing were available online from Blackwell Synergy until they were integrated into Wiley InterScience on June 30,2008, in December 2007, Wiley also began distributing its technical titles through the Safari Books Online e-reference service. On February 17,2012, Wiley announced the acquisition of Inscape Holdings Inc. which provides DISC assessments and training for interpersonal business skills. On August 13,2012, Wiley announced it entered into an agreement to sell all of its travel assets, including all of its interests in the Frommers brand. On October 2,2012, Wiley announced it would acquire Deltak edu, LLC, Deltak is expected to contribute solid growth to both Wileys Global Education business and Wiley overall. Seventh-generation members Jesse and Nate Wiley work in the companys Professional/Trade and Scientific, Technical, Medical, and Scholarly businesses, respectively. Wiley has been owned since 1962, and listed on the New York Stock Exchange since 1995, its stock is traded under the symbols NYSE, JW. A and NYSE
6.
Inverse Gaussian distribution
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In probability theory, the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on. Its probability density function is given by f =1 /2 exp for x >0, as λ tends to infinity, the inverse Gaussian distribution becomes more like a normal distribution. The inverse Gaussian distribution has several properties analogous to a Gaussian distribution and its cumulant generating function is the inverse of the cumulant generating function of a Gaussian random variable. To indicate that a random variable X is inverse Gaussian-distributed with mean μ, the probability density function of inverse Gaussian distribution has a single parameter form given by f = μ2 π x 3 /2 exp . In this form, the mean and variance of the distribution are equal, μ = σ2. Where z 1 = μ x 1 /2 − x 1 /2 and z 2 = μ x 1 /2 + x 1 /2, where the Φ is the cdf of standard normal distribution. The variables z 1 and z 2 are related to other by the identity z 22 = z 12 +4 μ. In the single form, the MGF simplifies to M = exp . An inverse Gaussian distribution in double parameter form f can be transformed into a single parameter form f by appropriate scaling y = μ2 x λ, the standard form of inverse Gaussian distribution is f =12 π x 3 /2 exp . If Xi has an IG distribution for i =1,2, N and all Xi are independent, then S = ∑ i =1 n X i ∼ I G. Note that Var E = μ02 w i 2 λ0 w i 2 = μ02 λ0 is constant for all i and this is a necessary condition for the summation. Otherwise S would not be inverse Gaussian, for any t >0 it holds that X ∼ I G ⇒ t X ∼ I G. The inverse Gaussian distribution is an exponential family with natural parameters -λ/ and -λ/2. The stochastic process Xt given by X0 =0 X t = ν t + σ W t is a Brownian motion with drift ν. Then, the first passage time for a fixed level α >0 by Xt is distributed according to an inverse-gaussian, a common special case of the above arises when the Brownian motion has no drift. In that case, parameter μ tends to infinity, and the first passage time for fixed level α has probability density function f = α σ2 π x 3 exp and this is a Lévy distribution with parameters c = α2 σ2 and μ =0. The model where X i ∼ I G, i =1,2, …, n with all wi known, unknown, μ ^ and λ ^ are independent and μ ^ ∼ I G n λ ^ ∼1 λ χ n −12. The following algorithm may be used, generate a random variate from a normal distribution with a mean of 0 and 1 standard deviation ν = N
7.
Ole Barndorff-Nielsen
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Ole Eiler Barndorff-Nielsen is a Danish statistician who has contributed to many areas of statistical science. He became interested in statistics when, as a student of actuarial mathematics and he graduated from the University of Aarhus in 1960, where he has spent most of his academic life, and where he became professor of statistics in 1973. Among Barndorff-Nielsens early scientific contributions are his work on exponential families and on the foundations of statistics, in particular sufficiency, in 1977 he introduced the hyperbolic distribution as a mathematical model of the size distribution of sand grains, formalising heuristic ideas proposed by Ralph Alger Bagnold. He also derived the larger class of generalised hyperbolic distributions and these distributions, in particular the normal-inverse Gaussian distribution, have later turned out to be useful in many other areas of science, in particular turbulence and finance. The NIG-distribution is now used to describe the distribution of returns from financial assets. In 1984 he produced a film on the physics of blown sand. Later Barndorff-Nielsen played a role in the application of differential geometry to investigate statistical models. He has jointly with David Cox written two books on asymptotic techniques in statistics. Barndorff-Nielsen is a member of the Royal Danish Academy of Sciences and Letters and he has received honorary doctorate degrees from the Université Paul Sabatier, Toulouse and the Katholieke Universiteit Leuven. In 1993-1995 he was a very influential president of the Bernoulli Society for Mathematical Statistics and he was the editor of International Statistical Review in 1980-1987 and of the journal Bernoulli in 1994-2000. In 2001 he received the Humboldt Prize and in 2010 the Faculty Price from Faculty of Science, generalised hyperbolic distribution Generalized inverse Gaussian distribution Hyperbolic distribution Information geometry List of mathematicians Barndorff-Nielsen, Ole. Information and exponential families in statistical theory, Wiley Series in Probability and Mathematical Statistics. Chichester, John Wiley \& Sons, Ltd. pp. ix+238 pp. ISBN 0-471-99545-2, the Centre for Mathematical Physics and Stochastics
8.
Georges Henri Halphen
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Georges Henri Halphen was a French mathematician. He did his studies at École Polytechnique and he was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry. He also worked on invariant theory and projective differential geometry and he received in the Steiner prize of the Prussian Academy of Sciences in 1880 along with Max Noether. He received the prix Poncelet in 1883 and the prix dOrmoy in 1885 and he was elected to the Académie des sciences in 1886 in the Section de Géométrie, replacing the deceased Jean-Claude Bouquet. In 1887 Halphen was elected to the Accademia dei Lincei in Rome, in 1872 he married Rose Marguerite Aron, with whom he had eight children, four sons and four daughters. Of the four sons, three joined the military and two of them died in World War I, one of his grandsons was Étienne Halphen, who did significant work in applied statistics. 1886,1888,1891 Bernkopf, Dictionary of Scientific Biography Gruson, Un aperçu des travaux mathématiques de G H Halphen in Complex projective geometry, London Math. Bézouts theorem Cramers paradox OConnor, John J. Robertson, Edmund F. Georges Henri Halphen, MacTutor History of Mathematics archive, University of St Andrews. Jewish Encyclopedia Biography on the Université de Rouen site A few of Halphens works available online Halphens dissertation and his 3 volume work on ellptic functions available online
9.
Gamma distribution
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In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The common exponential distribution and chi-squared distribution are special cases of the gamma distribution, there are three different parametrizations in common use, With a shape parameter k and a scale parameter θ. With a shape parameter α = k and a scale parameter β = 1/θ. With a shape parameter k and a mean parameter μ = k/β, in each of these three forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution for a random variable X for which E = kθ = α/β is fixed and greater than zero, and E = ψ + ln = ψ − ln is fixed. The parameterization with k and θ appears to be common in econometrics and certain other applied fields. For instance, in testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution. If k is an integer, then the distribution represents an Erlang distribution, i. e. the sum of k independent exponentially distributed random variables. The gamma distribution can be parameterized in terms of a shape parameter α = k, both parametrizations are common because either can be more convenient depending on the situation. The cumulative distribution function is the gamma function, F = ∫0 x f d u = γ Γ where γ is the lower incomplete gamma function. If α is an integer, the cumulative distribution function has the following series expansion. E − β x = e − β x ∑ i = α ∞ i i, here Γ is the gamma function evaluated at k. The cumulative distribution function is the gamma function, F = ∫0 x f d u = γ Γ where γ is the lower incomplete gamma function. It can also be expressed as follows, if k is a positive integer, I e − x / θ = e − x / θ ∑ i = k ∞1 i. I The skewness is equal to 2 / k, it only on the shape parameter. Unlike the mode and the mean which have readily calculable formulas based on the parameters, the median for this distribution is defined as the value ν such that 1 Γ θ k ∫0 ν x k −1 e − x / θ d x =12. A formula for approximating the median for any distribution, when the mean is known, has been derived based on the fact that the ratio μ/ is approximately a linear function of k when k ≥1. The approximation formula is ν ≈ μ3 k −0.83 k +0.2, K. P. Later, it was shown that λ is a convex function of m
10.
Zipf's law
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The law is named after the American linguist George Kingsley Zipf, who popularized it and sought to explain it, though he did not claim to have originated it. The French stenographer Jean-Baptiste Estoup appears to have noticed the regularity before Zipf and it was also noted in 1913 by German physicist Felix Auerbach. Zipfs law states that given some corpus of natural language utterances, for example, in the Brown Corpus of American English text, the word the is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences. True to Zipfs Law, the word of accounts for slightly over 3. 5% of words, followed by. Only 135 vocabulary items are needed to account for half the Brown Corpus, the appearance of the distribution in rankings of cities by population was first noticed by Felix Auerbach in 1913. When Zipfs law is checked for cities, a better fit has been found with exponent s =1.07, while Zipfs law holds for the upper tail of the distribution, the entire distribution of cities is log-normal and follows Gibrats law. Both laws are consistent because a log-normal tail can not be distinguished from a Pareto tail. Zipfs law is most easily observed by plotting the data on a log-log graph, for example, the word the would appear at x = log, y = log. It is also possible to plot reciprocal rank against frequency or reciprocal frequency or interword interval against rank, the data conform to Zipfs law to the extent that the plot is linear. Formally, let, N be the number of elements, k be their rank and it has been claimed that this representation of Zipfs law is more suitable for statistical testing, and in this way it has been analyzed in more than 30,000 English texts. The goodness-of-fit tests yield that only about 15% of the texts are statistically compatible with this form of Zipfs law, slight variations in the definition of Zipfs law can increase this percentage up to close to 50%. In the example of the frequency of words in the English language, N is the number of words in the English language and, if we use the version of Zipfs law. F will then be the fraction of the time the kth most common word occurs, the law may also be written, f =1 k s H N, s where HN, s is the Nth generalized harmonic number. The simplest case of Zipfs law is a 1⁄f function, given a set of Zipfian distributed frequencies, sorted from most common to least common, the second most common frequency will occur ½ as often as the first. The third most common frequency will occur ⅓ as often as the first, the fourth most common frequency will occur ¼ as often as the first. The nth most common frequency will occur 1⁄n as often as the first, however, this cannot hold exactly, because items must occur an integer number of times, there cannot be 2.5 occurrences of a word. Nevertheless, over fairly wide ranges, and to a good approximation. Mathematically, the sum of all frequencies in a Zipf distribution is equal to the harmonic series