Context: Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion., Johann Peter Gustav Lejeune Dirichlet (or ; 13 February 1805  5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function., (X_i) = n p_i (1-p_i)</math><br>formula_1|
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for rolling a "k" sided dice "n" times. For "n" independent trials each of which leads to a success for exactly one of "k" categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories., Mathematical analysis is the branch of mathematics dealing with limits
and related theories, such as differentiation, integration, measure, infinite series, and analytic functions., George Alfred Barnard (23 September 1915  9 August 2002) was a British statistician known particularly for his work on the foundations of statistics and on quality control., In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet "L"-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
Branches of analytic number theory.
Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique., Robert O. Schlaifer (13 September 1914  24 July 1994) was a pioneer of Bayesian decision theory. At the time of his death he was William Ziegler Professor of Business Administration Emeritus of the Harvard Business School. In 1961 he was elected as a Fellow of the American Statistical Association., Thomas Bayes (c. 1701 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for having formulated a specific case of the theorem that bears his name: Bayes' theorem. Bayes never published what would eventually become his most famous accomplishment; his notes were edited and published after his death by Richard Price., </math> In probability theory and statistics, the binomial distribution with parameters "n" and "p" is the discrete probability distribution of the number of successes in a sequence of "n" independent yes/no experiments, each of which yields success with probability "p". A success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when "n" = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance., Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers)., Debt, AIDS, Trade, Africa (DATA) was a multinational non-government organization founded in January 2002 in London by U2's Bono along with Bobby Shriver and activists from the Jubilee 2000 Drop the Debt campaign., In Bayesian probability theory, if the posterior distributions "p"(|"x") are in the same family as the prior probability distribution "p"(), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function. For example, the Gaussian family is conjugate to itself (or "self-conjugate") with respect to a Gaussian likelihood function: if the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian. This means that the Gaussian distribution is a conjugate prior for the likelihood that is also Gaussian. The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory. A similar concept had been discovered independently by George Alfred Barnard., Bayesian statistics, named for Thomas Bayes (17011761), is a theory in the field of statistics in which the evidence about the true state of the world is expressed in terms of "degrees of belief" known as Bayesian probabilities. Such an interpretation is only one of a number of interpretations of probability and there are other statistical techniques that are not based on 'degrees of belief'. One of the key ideas of Bayesian statistics is that "probability is orderly opinion, and that inference from data is nothing other than the revision of such opinion in the light of relevant new information.", In probability theory and statistics, a probability distribution is a mathematical function that, stated in simple terms, can be thought of as providing the probability of occurrence of different possible outcomes in an experiment. For instance, if the random variable X is used to denote the outcome of a coin toss ('the experiment'), then the probability distribution of X would take the value 0.5 for formula_1, and 0.5 for formula_2. , An "a priori" probability is a probability that is derived purely by deductive reasoning. One way of deriving "a priori" probabilities is the principle of indifference, which has the character of saying that, if there are "N" mutually exclusive and exhaustive events and if they are equally likely, then the probability of a given event occurring is 1/"N". Similarly the probability of one of a given collection of "K" events is "K"/"N"., ([x=i]) = p_i (1-p_i)</math><br>formula_1|, Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384322 B.C.), though "analysis" as a formal concept is a relatively recent development., In mathematics, a Fourier series is a way to represent a (wave-like) function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1. Fourier series are also central to the original proof of the NyquistShannon sampling theorem. The study of Fourier series is a branch of Fourier analysis., In probability and statistics , the Dirichlet distribution ( after Peter Gustav Lejeune Dirichlet ) , often denoted \ operatorname ( Dir ) ( \ boldsymbol \ alpha ) , is a family of continuous multivariate probability distributions parameterized by a vector \ boldsymbol \ alpha of positive reals . It is the multivariate generalization of the beta distribution . Dirichlet distributions are very often used as prior distributions in Bayesian statistics , and in fact the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution . The infinite - dimensional generalization of the Dirichlet distribution is the Dirichlet process ., Howard Raiffa (January 24, 1924  July 8, 2016) was an American academic who was the Frank P. Ramsey Professor (Emeritus) of Managerial Economics, a joint chair held by the Business School and the Kennedy School of Government at Harvard University. He was an influential Bayesian decision theorist and pioneer in the field of decision analysis, with works in statistical decision theory, game theory, behavioral decision theory, risk analysis, and negotiation analysis. He helped found and was the first director of the International Institute for Applied Systems Analysis. , Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. 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 and experiments., In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, denoted by "" and "", that appear as exponents of the random variable and control the shape of the distribution., Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, assigned probabilities represent states of knowledge or belief., In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions., A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems., Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty). The higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is unbiased, the two outcomes ("head" and "tail") are both equally probable; the probability of "head" equals the probability of "tail". Since no other outcomes are possible, the probability is 1/2 (or 50%), of either "head" or "tail". In other words, the probability of "head" is 1 out of 2 outcomes and the probability of "tail" is also 1 out of 2 outcomes, expressed as 0.5 when converted to decimal, with the above-mentioned quantification system. This type of probability is also called a priori probability., Subject: dirichlet distribution, Relation: named_after, Options: (A) 1 (B) 2 (C) 5 (D) 7 (E) 8 (F) 9 (G) africa (H) aristotle (I) c (J) concept (K) death (L) english (M) family (N) johann peter gustav lejeune dirichlet (O) london (P) mathematics (Q) p (R) peter (S) robert (T) technique (U) thomas bayes (V) wave
Output: johann peter gustav lejeune dirichlet