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Poisson Binomial Distribution
In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson. In other words, it is the probability distribution of the number of successes in a collection of ''n'' independent yes/no experiments with success probabilities p_1, p_2, \dots , p_n. The ordinary binomial distribution is a special case of the Poisson binomial distribution, when all success probabilities are the same, that is p_1 = p_2 = \cdots = p_n. Definitions Probability Mass Function The probability of having ''k'' successful trials out of a total of ''n'' can be written as the sum :\Pr(K=k) = \sum\limits_ \prod\limits_ p_i \prod\limits_ (1-p_j) where F_k is the set of all subsets of ''k'' integers that can be selected from . For example, if ''n'' = 3, then F_2=\left\. A^c is the complement of A, i.e. A^c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory 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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divide-and-conquer Algorithm
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform (FFT). Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational cost is o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Le Cam's Theorem
In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following. Suppose: * X_1, X_2, X_3, \ldots are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed. * \Pr(X_i = 1) = p_i, \text i = 1, 2, 3, \ldots. * \lambda_n = p_1 + \cdots + p_n. * S_n = X_1 + \cdots + X_n. (i.e. S_n follows a Poisson binomial distribution) Then :\sum_^\infty \left, \Pr(S_n=k) - \ < 2 \left( \sum_^n p_i^2 \right). In other words, the sum has approximately a and the above inequality bounds the approximation error in terms of the |
Binomial Distribution
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 experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: ''success'' (with probability ''p'') or ''failure'' (with probability q=1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., ''n'' = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size ''n'' drawn with replacement from a population of size ''N''. If the sampling is carried out without replacement, the draws are not independent and so the resulting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ingram Olkin
Ingram Olkin (July 23, 1924 – April 28, 2016) was a professor emeritus and chair of statistics and education at Stanford University and the Stanford Graduate School of Education. He is known for developing statistical analysis for evaluating policies, particularly in education, and for his contributions to meta-analysis, statistics education, multivariate analysis, and majorization theory. Biography Olkin was born in 1924 in Waterbury, Connecticut. He received a B.S. in mathematics at the City College of New York, an M.A. from Columbia University, and his Ph.D. from the University of North Carolina. Olkin also studied with Harold Hotelling. Olkin's advisor was S. N. Roy and his Ph.D. thesis was "On distribution problems in multivariate analysis" submitted in 1951. Olkin died from complications of colorectal cancer at his home in Palo Alto, California on April 28, 2016, aged 91. A spokesperson for the statistics profession: Honors and awards Olkin was awarded the fourth bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence Shepp
Lawrence Alan Shepp (September 9, 1936 Brooklyn, NY – April 23, 2013, Tucson, AZ) was an American mathematician, specializing in statistics and computational tomography. Shepp obtained his PhD from Princeton University in 1961 with a dissertation entitled ''Recurrent Sums of Random Variables''. His advisor was William Feller. He joined Bell Laboratories in 1962. He joined Rutgers University in 1997. He joined University of Pennsylvania in 2010. His work in tomography has had biomedical imaging applications, and he has also worked as professor of radiology at Columbia University (1973–1996), as a mathematician in the radiology service of Columbia Presbyterian Hospital. Awards and honors * 2014: IEEE Marie Sklodowska-Curie Award * 2012: Became a fellow of the American Mathematical Society. * 1992: Elected member of the Institute of Medicine * 1989: Elected member of the National Academy of Sciences * 1979: IEEE Distinguished Scientist Award in 1979 * 1979: Lester R. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and Statistical independence, independently of the time since the last event. It is named after France, French mathematician Siméon Denis Poisson (; ). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. The DFT is the most important discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution Of Probability Distributions
The convolution/sum of probability distributions arises in probability theory and statistics as the operation in terms of probability distributions that corresponds to the addition of independent random variables and, by extension, to forming linear combinations of random variables. The operation here is a special case of convolution in the context of probability distributions. Introduction The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions: see List of convolutions of probability distributions The general formula for the distribution of the sum Z=X+Y of two independent integer-valued (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerically Stable
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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