1.
Beta distribution
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The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. In Bayesian inference, the distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial. The beta distribution is a model for the random behavior of percentages. The beta function, B, is a constant to ensure that the total probability integrates to 1. In the above equations x is an observed value that actually occurred—of a random process X. L. Johnson. Several authors, including N. L. Johnson and S, the probability density function satisfies the differential equation f ′ = f x − x. The cumulative distribution function is F = B B = I x where B is the beta function. The mode of a Beta distributed random variable X with α, β >1 is the most likely value of the distribution, when both parameters are less than one, this is the anti-mode, the lowest point of the probability density curve. Letting α = β, the expression for the mode simplifies to 1/2, showing that for α = β >1 the mode, is at the center of the distribution, it is symmetric in those cases. See Shapes section in this article for a full list of mode cases, for several of these cases, the maximum value of the density function occurs at one or both ends. In some cases the value of the density function occurring at the end is finite, for example, in the case of α =2, β =1, the density function becomes a right-triangle distribution which is finite at both ends. In several other cases there is a singularity at one end, for example, in the case α = β = 1/2, the Beta distribution simplifies to become the arcsine distribution. There is debate among mathematicians about some of cases and whether the ends can be called modes or not. There is no general closed-form expression for the median of the distribution for arbitrary values of α and β. Closed-form expressions for particular values of the parameters α and β follow, For symmetric cases α = β, median = 1/2. For α =1 and β >0, median =1 −2 −1 β For α >0 and β =1, median =2 −1 α For α =3 and β =2, median =0.6142724318676105. The real solution to the quartic equation 1 − 8x3 + 6x4 =0, for α =2 and β =3, median =0.38572756813238945. When α, β ≥1, the error in this approximation is less than 4%
2.
Saxony
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Its capital is Dresden, and its largest city is Leipzig. Saxony is the tenth largest of Germanys sixteen states, with an area of 18,413 square kilometres, located in the middle of a large, formerly all German-speaking part of Europe, the history of the state of Saxony spans more than a millennium. It has been a medieval duchy, an electorate of the Holy Roman Empire, a kingdom, the area of the modern state of Saxony should not be confused with Old Saxony, the area inhabited by Saxons. Old Saxony corresponds approximately to the modern German states of Lower Saxony, Saxony-Anhalt, Saxony is divided into 10 districts,1. After a reform in 2008, these regions - with some alterations of their respective areas - were called Direktionsbezirke, in 2012, the authorities of these regions were merged into one central authority, the Landesdirektion Sachsen. The Erzgebirgskreis district includes the Ore Mountains, and the Schweiz-Osterzgebirge district includes Saxon Switzerland, the largest cities in Saxony according to the 31 December 2015 estimate. To this can be added that Leipzig forms a metropolitan region with Halle. The latter city is located just across the border to Saxony-Anhalt, Leipzig shares for instance an S-train system and an airport with Halle. Saxony has, after Saxony Anhalt, the most vibrant economy of the states of the former East Germany and its economy grew by 1. 9% in 2010. Nonetheless, unemployment remains above the German average, the eastern part of Germany, excluding Berlin, qualifies as an Objective 1 development-region within the European Union, and is eligible to receive investment subsidies of up to 30% until 2013. FutureSAX, a business competition and entrepreneurial support organisation, has been in operation since 2002. Microchip makers near Dresden have given the region the nickname Silicon Saxony, the publishing and porcelain industries of the region are well known, although their contributions to the regional economy are no longer significant. Today the automobile industry, machinery production and services contribute to the development of the region. Saxony is also one of the most renowned tourist destinations in Germany - especially the cities of Leipzig and Dresden, new tourist destinations are developing, notably in the lake district of Lausitz. Saxony reported an unemployment of 8. 8% in 2014. By comparison the average in the former GDR was 9. 8% and 6. 7% for Germany overall, the unemployment rate reached 8. 2% in May 2015. The Leipzig area, which recently was among the regions with the highest unemployment rate, could benefit greatly from investments by Porsche. With the VW Phaeton factory in Dresden, and many part suppliers, zwickau is another major Volkswagen location
3.
Mammal
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Mammals are any vertebrates within the class Mammalia, a clade of endothermic amniotes distinguished from reptiles by the possession of a neocortex, hair, three middle ear bones and mammary glands. All female mammals nurse their young with milk, secreted from the mammary glands, Mammals include the largest animals on the planet, the great whales. The basic body type is a quadruped, but some mammals are adapted for life at sea, in the air, in trees. The largest group of mammals, the placentals, have a placenta, Mammals range in size from the 30–40 mm bumblebee bat to the 30-meter blue whale. With the exception of the five species of monotreme, all modern mammals give birth to live young, most mammals, including the six most species-rich orders, belong to the placental group. The largest orders are the rodents, bats and Soricomorpha, the next three biggest orders, depending on the biological classification scheme used, are the Primates, the Cetartiodactyla, and the Carnivora. Living mammals are divided into the Yinotheria and Theriiformes There are around 5450 species of mammal, in some classifications, extant mammals are divided into two subclasses, the Prototheria, that is, the order Monotremata, and the Theria, or the infraclasses Metatheria and Eutheria. The marsupials constitute the group of the Metatheria, and include all living metatherians as well as many extinct ones. Much of the changes reflect the advances of cladistic analysis and molecular genetics, findings from molecular genetics, for example, have prompted adopting new groups, such as the Afrotheria, and abandoning traditional groups, such as the Insectivora. The mammals represent the only living Synapsida, which together with the Sauropsida form the Amniota clade, the early synapsid mammalian ancestors were sphenacodont pelycosaurs, a group that produced the non-mammalian Dimetrodon. At the end of the Carboniferous period, this group diverged from the line that led to todays reptiles. Some mammals are intelligent, with some possessing large brains, self-awareness, Mammals can communicate and vocalize in several different ways, including the production of ultrasound, scent-marking, alarm signals, singing, and echolocation. Mammals can organize themselves into fission-fusion societies, harems, and hierarchies, most mammals are polygynous, but some can be monogamous or polyandrous. They provided, and continue to provide, power for transport and agriculture, as well as commodities such as meat, dairy products, wool. Mammals are hunted or raced for sport, and are used as model organisms in science, Mammals have been depicted in art since Palaeolithic times, and appear in literature, film, mythology, and religion. Defaunation of mammals is primarily driven by anthropogenic factors, such as poaching and habitat destruction, Mammal classification has been through several iterations since Carl Linnaeus initially defined the class. No classification system is accepted, McKenna & Bell and Wilson & Reader provide useful recent compendiums. Though field work gradually made Simpsons classification outdated, it remains the closest thing to a classification of mammals
4.
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
5.
Kurtosis
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In probability theory and statistics, kurtosis is a measure of the tailedness of the probability distribution of a real-valued random variable. Depending on the measure of kurtosis that is used, there are various interpretations of kurtosis. The standard measure of kurtosis, originating with Karl Pearson, is based on a version of the fourth moment of the data or population. This number is related to the tails of the distribution, not its peak, hence, for this measure, higher kurtosis is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations. The kurtosis of any normal distribution is 3. It is common to compare the kurtosis of a distribution to this value, distributions with kurtosis less than 3 are said to be platykurtic, although this does not imply the distribution is flat-topped as sometimes reported. Rather, it means the distribution produces fewer and less extreme outliers than does the normal distribution, an example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with kurtosis greater than 3 are said to be leptokurtic and it is also common practice to use an adjusted version of Pearsons kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the normal distribution. Some authors use kurtosis by itself to refer to the excess kurtosis, for the reason of clarity and generality, however, this article follows the non-excess convention and explicitly indicates where excess kurtosis is meant. Alternative measures of kurtosis are, the L-kurtosis, which is a version of the fourth L-moment. These are analogous to the measures of skewness that are not based on ordinary moments. The kurtosis is the fourth standardized moment, defined as Kurt = μ4 σ4 = E 2, several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant, other choices include γ2, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis. The kurtosis is bounded below by the squared skewness plus 1, μ4 σ4 ≥2 +1, the lower bound is realized by the Bernoulli distribution. There is no limit to the excess kurtosis of a general probability distribution. A reason why some authors favor the excess kurtosis is that cumulants are extensive, formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. Xn be independent random variables for which the fourth moment exists, the excess kurtosis of Y is Kurt −3 =12 ∑ i =1 n σ i 4 ⋅, where σ i is the standard deviation of X i. In particular if all of the Xi have the same variance, the reason not to subtract off 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed
6.
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
7.
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
8.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
9.
Benford's law
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Benfords law, also called the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in naturally occurring collections of numbers. For example, in sets which obey the law, the number 1 appears as the most significant digit about 30% of the time, by contrast, if the digits were distributed uniformly, they would each occur about 11. 1% of the time. Benfords law also makes predictions about the distribution of digits, third digits, digit combinations. It tends to be most accurate values are distributed across multiple orders of magnitude. The graph here shows Benfords law for base 10, there is a generalization of the law to numbers expressed in other bases, and also a generalization from leading 1 digit to leading n digits. It is named after physicist Frank Benford, who stated it in 1938, Benfords law is a special case of Zipfs law. A set of numbers is said to satisfy Benfords law if the digit d occurs with probability P = log 10 − log 10 = log 10 = log 10 . Therefore, this is the distribution expected if the mantissae of the logarithms of the numbers are uniformly and randomly distributed. For example, a x, constrained to lie between 1 and 10, starts with the digit 1 if 1 ≤ x <2. Therefore, x starts with the digit 1 if log 1 ≤ log x < log 2, the probabilities are proportional to the interval widths, and this gives the equation above. An extension of Benfords law predicts the distribution of first digits in other bases besides decimal, in fact, the general form is, P = log b − log b = log b . For b =2, Benfords law is true but trivial, the discovery of Benfords law goes back to 1881, when the American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages were much more worn than the other pages. Newcombs published result is the first known instance of this observation and includes a distribution on the second digit, Newcomb proposed a law that the probability of a single number N being the first digit of a number was equal to log − log. The phenomenon was noted in 1938 by the physicist Frank Benford. The total number of used in the paper was 20,229. This discovery was named after Benford. In 1995, Ted Hill proved the result about mixed distributions mentioned below, arno Berger and Ted Hill have stated that, The widely known phenomenon called Benford’s law continues to defy attempts at an easy derivation