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Galton Board
The Galton board, also known as the Galton box or quincunx or bean machine, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximates a normal distribution. Among its applications, it afforded insight into regression to the mean or "reversion to mediocrity". Description The Galton board consists of a vertical board with interleaved rows of pegs. Beads are dropped from the top and, when the device is level, bounce either left or right as they hit the pegs. Eventually they are collected into bins at the bottom, where the height of bead columns accumulated in the bins approximate a bell curve. Overlaying Pascal's triangle onto the pins shows the number of different paths that can be taken to get to each bin. Large-scale working models of this device created by Charles and Ray Eames can be seen in the '' Mathematica: A World of Numbers... and Beyond'' exhibits permane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galton Box
The Galton board, also known as the Galton box or quincunx or bean machine, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximates a normal distribution. Among its applications, it afforded insight into Regression toward the mean, regression to the mean or "reversion to mediocrity". Description The Galton board consists of a vertical board with interleaved rows of pegs. Beads are dropped from the top and, when the device is level, bounce either left or right as they hit the pegs. Eventually they are collected into bins at the bottom, where the height of bead columns accumulated in the bins approximate a normal distribution, bell curve. Overlaying Pascal's triangle onto the pins shows the number of different paths that can be taken to get to each bin. Large-scale working models of this device created by Charles and Ray Eames can be seen in the ''Mathematica: A Worl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log-normal Distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal distribution. Equivalently, if has a normal distribution, then the exponential function of , , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics). The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas. A log-normal process is the statistical realization of the multipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Payazzo
Payazzo (or pajatso) is a traditional Finnish gambling arcade game, dating back to the 1920s, when it was introduced into Finland from Germany. The object of payazzo is to flick a coin into one of the winning slots. When the attempt is successful, the machine rewards the player with a couple of coins. If the attempt is unsuccessful, the player loses the flicked coin. Name The game is called ''pajatso'' in Finnish. It is a Fennicized form of ''bajazzo'', the name of the early German models. ''Bajazzo'' refers to an Italian-style clown ('' pagliaccio'' in Italian). Finland's Slot Machine Association (Finnish: ''Raha-automaattiyhdistys'' or ''RAY''), the manufacturer of today's payazzo machines, uses ''payazzo'' in English for ''pajatso''. In Finland, the game is often referred to informally as ''Jasso''. The popularity of the nickname was proven as the manufacturer decided to use only the informal form in the names of some models of payazzo combining coin flicking with an electr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pachinko
is a mechanical game originating in Japan that is used as an arcade game, and much more frequently for gambling. Pachinko fills a niche in Japanese gambling comparable to that of the slot machine in the West as a form of low-stakes, low-strategy gambling. Pachinko parlors are widespread in Japan, and usually also feature a number of slot machines (called ''pachislo'' or pachislots) so these venues look and operate similarly to casinos. Modern pachinko machines have both mechanical and digital components. Gambling for cash is illegal in Japan, but the widespread popularity of low-stakes pachinko in Japanese society has enabled a specific legal loophole allowing it to exist. Pachinko balls won from games cannot be exchanged directly for money in the parlor, nor can they be removed from the premises or exchanged with other parlors. However, they can be legally traded to the parlor for so-called "special prize" tokens (特殊景品 ''tokushu keihin''), which can in turn be "so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bagatelle
Bagatelle (from the Château de Bagatelle) is a billiards-derived indoor table game, the object of which is to get a number of balls (set at nine in the 19th century) past wooden pins (which act as obstacles) into holes that are guarded by wooden pegs; penalties are incurred if the pegs are knocked over. It probably developed from the table made with raised sides for ''trou madame'', which was also played with ivory balls and continued to be popular into the later 19th century, after which it developed into bar billiards, with influences from the French/Belgian game ' (with supposed Russian origins). A bagatelle variant using fixed metal pins, ''billard japonais'', eventually led to the development of pachinko and pinball. History Table games involving sticks and balls evolved from efforts to bring outdoor games like ground billiards, croquet, and bowling inside for play during inclement weather. They are attested in general by the 15th century, although the 19th-century idea tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sir Francis Galton
Sir Francis Galton, FRS FRAI (; 16 February 1822 – 17 January 1911), was an English Victorian era polymath: a statistician, sociologist, psychologist, anthropologist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician and a proponent of social Darwinism, eugenics, and scientific racism. He was knighted in 1909. Galton produced over 340 papers and books. He also created the statistical concept of correlation and widely promoted regression toward the mean. He was the first to apply statistical methods to the study of human differences and inheritance of intelligence, and introduced the use of questionnaires and surveys for collecting data on human communities, which he needed for genealogical and biographical works and for his anthropometric studies. He was a pioneer of eugenics, coining the term itself in 1883, and also coined the phrase " nature versus nurture". His book ''Hereditary Genius'' (1869) was the first social sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Entropy Probability Distribution
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class (usually defined in terms of specified properties or measures), then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time. Definition of entropy and differential entropy If X is a discrete random variable with distribution given by :\operatorname(X=x_k) = p_k \quad\mbox k=1,2,\ldots then the entropy of X is defined as :H(X) = - \sum_p_k\log p_k . If X is a continuous random variable with probabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. It has found far-ranging applications in chemistry and physics, in biological systems and their relation to life, in cosmology, economics, sociology, weather science, climate change, and information systems including the transmission of information in telecommunication. The thermodynamic concept was referred to by Scottish scientist and engineer William Rankine in 1850 with the names ''thermodynamic function'' and ''heat-potential''. In 1865, German physicist Rudolf Clausius, one of the leading founders of the field of thermodynamics, defined it as the quotient of an infinitesimal amount of hea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Deviations
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter '' s'', for the sample standard deviation. The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data. The standard deviation of a popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Moivre–Laplace Theorem
In probability theory, the de Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. In particular, the theorem shows that the probability mass function of the random number of "successes" observed in a series of n statistical independence, independent Bernoulli trials, each having probability p of success (a binomial distribution with n trials), Convergence in distribution, converges to the probability density function of the normal distribution with mean np and standard deviation \sqrt, as n grows large, assuming p is not 0 or 1. The theorem appeared in the second edition of ''The Doctrine of Chances'' by Abraham de Moivre, published in 1738. Although de Moivre did not use the term "Bernoulli trials", he wrote about the probability distribution of the number of times "heads" appears when a coin is tossed 3600 times. This is one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Aitchison
John Aitchison (22 July 1926 – 23 December 2016) was a Scottish statistician. Career John Aitchison studied at the Universitiy of Edinburgh after being uncomfortable explaining to his headmaster that he didn’t plan to attend university. He graduated in 1947 with an MA in mathematics. After two years wherein he did actuarial work, he also attended Trinity College, Cambridge. He had a scholarship to do so, and graduated in 1951 with a BA focused on statistics. The year after he graduated, he joined the Department of Applied Economics at Cambridge as a statistician. He continued his work at Cambridge until 1956, when he was offered the position of Lecturer of Statistics at the University of Glasgow. During his time at Glasgow, he wrote ''The lognormal distribution, with special reference to its uses in economics (1957)'' with J A C Brown (who he met at Cambridge). However, he left Glasgow in 1962, when the University of Liverpool offered him the positions of Senior Lect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
_-_Galton_1889_diagram.png "Click for more on -> De Moivre–Laplace Theorem")