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Poisson-Dirichlet Distribution
In probability theory, Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-increasing sequences with sum 1, depending on two parameters \alpha \in [0,1) and \theta \in (-\alpha, \infty). It can be defined as follows. One considers independent random variables (Y_n)_ such that Y_n follows the beta distribution of parameters 1-\alpha and \theta+n \alpha. Then, the Poisson-Dirichlet distribution PD(\alpha, \theta) of parameters \alpha and \theta is the law of the random decreasing sequence containing Y_1 and the products Y_n \prod_^(1-Y_k). This definition is due to Jim Pitman and Marc Yor. It generalizes Kingman's law, which corresponds to the particular case \alpha = 0. Number theory Patrick Billingsley has proven the following result: if n is a uniform random integer in \, if k \geq 1 is a fixed integer, and if p_1 \geq p_2 \geq \dots \geq p_k are the k largest prime divisors of n (with p_j arbitrarily defined if n has le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distributions
In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the variable and its complement to 1, respectively, and control the shape parameter, shape of the distribution. 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. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior distribution, conjugate prior probability distribution for the Bernoulli distribution, Bernoulli, binomial distribution, binomial, negative binomial distribution, negative binomial, and geometric distribution, geometric distributions. The formulation of the beta dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Pitman
Jim Pitman is a professor emeritus of statistics and mathematics at the University of California, Berkeley. Biography Jim Pitman (James W. Pitman) was born in Hobart, Australia, in June 1949, son of E. J. G. Pitman and Elinor J. Pitman, daughter of W. N. T. Hurst. He attended the Hutchins School, Hobart, Australia from 1954 to 1966, then the Australian National University (ANU) in Canberra, from 1967 to 1970. He received a BSc degree from the ANU in 1970, followed by a PhD in probability and statistics in 1974 from the University of Sheffield, with advisor Terry Speed. He lectured at the Universities of Copenhagen, Berkeley and Cambridge, from 1974 to 1978, before joining Berkeley as an assistant professor in 1978. Following promotion to professor in 1984, he retired from teaching duties at Berkeley in July 2021. He is now emeritus professor of statistics and mathematics at the University of California, Berkeley. Pitman is a Fellow of the Institute of Mathematical Statistics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Yor
Marc Yor (24 July 1949 – 9 January 2014) was a French mathematician well known for his work on stochastic processes, especially properties of semimartingales, Brownian motion and other Lévy processes, the Bessel processes, and their applications to mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that req .... Background Yor was a professor at the Paris VI University in Paris, France, from 1981 until his death in 2014. He was a recipient of several awards, including the Humboldt Prize, the Montyon Prize,Official biography at the French Academy website and was awarded t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patrick Billingsley
Patrick Paul Billingsley (May 3, 1925 – April 22, 2011) was an American mathematician and stage and screen actor, noted for his books in advanced probability theory and statistics. He was born and raised in Sioux Falls, South Dakota, and graduated from the United States Naval Academy in 1946. In '' Young Men and Fire'', fellow University of Chicago professor Norman Maclean wrote about Billingsley that "he is a distinguished statistician and one of the best amateur actors I have ever seen". Academic career After earning a Ph.D. in mathematics at Princeton University in 1955, he was attached to the NSA until his discharge from the Navy in 1957. In 1958 he became a professor of mathematics and statistics at the University of Chicago, where he served as chair of the Department of Statistics from 1980 to 1983, and retired in 1994. In 1964–65 he was a Fulbright Fellow and visiting professor at the University of Copenhagen. In 1971–72 he was a Guggenheim Fellow and visiting pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ewens's Sampling Formula
In population genetics, Ewens's sampling formula describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample. Definition Ewens's sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of ''n'' gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are ''a''1 alleles represented once in the sample, and ''a''2 alleles represented twice, and so on, is :\operatorname(a_1,\dots,a_n; \theta)=\prod_^n, for some positive number ''θ'' representing the population mutation rate, whenever a_1, \ldots, a_n is a sequence of nonnegative integers such that :a_1+2a_2+3a_3+\cdots+na_n=\sum_^ i a_i = n.\, The phrase "under certain conditions" used above is made precise by the following assumptions: * The sample size ''n'' is small by comparison to the size of the whole population; and * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warren Ewens
Warren John Ewens (born 23 January 1937 in Canberra) is an Australian-born mathematician who has been Professor of Biology at the University of Pennsylvania since 1997. (He also held that position 1972–1977.) He concentrates his research on the mathematical, statistical and theoretical aspects of population genetics. Ewens has worked in mathematical population genetics, computational biology, and evolutionary population genetics. He introduced Ewens's sampling formula. Ewens received a B.A. (1958) and M.A. (1960) in Mathematical Statistics from the University of Melbourne, where he was a resident student at Trinity College,"Salvete 1955", The Fleur-de-Lys', Nov. 1955, p. 14. and a Ph.D. from the Australian National University (1963) under P. A. P. Moran. He first joined the department of biology at the University of Pennsylvania in 1972, and in 2006 was named the Christopher H. Browne Distinguished Professor of Biology. Positions held include: *1967–1972 Foundati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
