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List Of Probability Distributions
Many probability distributions that are important in theory or applications have been given specific names. Discrete distributions With finite support * The Bernoulli distribution, which takes value 1 with probability ''p'' and value 0 with probability ''q'' = 1 − ''p''. * The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2. * The binomial distribution, which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success. * The beta-binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with heterogeneity in the success probability. * The degenerate distribution at ''x''0, where ''X'' is certain to take the value ''x''0. This does not look random, but it satisfies the definition of random variable. This is useful because it puts deterministic variables and random variables in the same formalism. * The disc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes 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). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benford's Law
Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore P. HillBenford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem 2011. In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on. The graph to the right shows Benford's law for base 10, one of infinitely many cases of a generalized law regarding numbers expressed in arbitrary (integer) bases, which rules out the possibility that the phenomenon might be an artifact of the base-10 number syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbs Distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain microstate (statistical mechanics), state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: :p_i \propto e^ where is the probability of the system being in state , is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol \propto denotes proportionality (mathematics), proportionality (see for the proportionality constant). The term ''system'' here has a very wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a Natural gas storage, natural gas storage tank. Therefore the Bolt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermal Equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in thermal equilibrium with itself if the temperature within the system is spatially uniform and temporally constant. Systems in thermodynamic equilibrium are always in thermal equilibrium, but the converse is not always true. If the connection between the systems allows transfer of energy as 'change in internal energy' but does not allow transfer of matter or transfer of energy as work, the two systems may reach thermal equilibrium without reaching thermodynamic equilibrium. Two varieties of thermal equilibrium Relation of thermal equilibrium between two thermally connected bodies The relation of thermal equilibrium is an instance of equilibrium between two bodies, which means that it refers to transfer through a selectively permeable p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develop the Phenomenology (particle physics), phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: :p_i \propto e^ where is the probability of the system being in state , is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol \propto denotes proportionality (see for the proportionality constant). The term ''system'' here has a very wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore the Boltzmann distribution can be used to solve a very wide variety of problems. The distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Negative Binomial Distribution
In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable X equal to the number of failures needed to get r successes in a sequence of independent Bernoulli trials. The probability p of success on each trial stays constant within any given experiment but varies across different experiments following a beta distribution. Thus the distribution is a compound probability distribution. This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distributionJohnson et al. (1993) or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution. If parameters of the beta distribution are \alpha and \beta, and if : X \mid p \sim \mathrm(r,p), where : p \sim \textrm(\alpha,\beta), then the marginal distribution of X is a beta negative binomial distribution: : X \sim \mathrm(r,\alpha,\beta). In the above, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skellam Distribution
The Skellam distribution is the discrete probability distribution of the difference N_1-N_2 of two statistically independent random variables N_1 and N_2, each Poisson-distributed with respective expected values \mu_1 and \mu_2. It is useful in describing the statistics of the difference of two images with simple photon noise, as well as describing the point spread distribution in sports where all scored points are equal, such as baseball, hockey and soccer. The distribution is also applicable to a special case of the difference of dependent Poisson random variables, but just the obvious case where the two variables have a common additive random contribution which is cancelled by the differencing: see Karlis & Ntzoufras (2003) for details and an application. The probability mass function for the Skellam distribution for a difference K=N_1-N_2 between two independent Poisson-distributed random variables with means \mu_1 and \mu_2 is given by: : p(k;\mu_1,\mu_2) = \Pr\ = e^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Pmf
Poisson may refer to: People *Siméon Denis Poisson, French mathematician Places *Poissons, a commune of Haute-Marne, France *Poisson, Saône-et-Loire, a commune of Saône-et-Loire, France Other uses *Poisson (surname), a French surname *Poisson (crater), a lunar crater named after Siméon Denis Poisson *The French word for fish See also *Adolphe-Poisson Bay, a body of water located to the southwest of Gouin Reservoir, in La Tuque, Mauricie, Quebec *Poisson distribution, a discrete probability distribution named after Siméon Denis Poisson *Poisson's equation, a partial differential equation named after Siméon Denis Poisson *List of things named after Siméon Denis Poisson These are things named after Siméon Denis Poisson (1781 – 1840), a French mathematician. Physics * ''Poisson’s Equations'' (thermodynamics) * ''Poisson’s Equation'' (rotational motion) * Schrödinger–Poisson equation * Vlasov–Poisson equ ... * Poison (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CMP PMF
CMP may refer to: Medicine * Cardiomyopathy, a heart muscle disease * Chondromalacia patellae, a degenerative condition of the knee cap (patella) * Chronic myofascial pain, also known as myofascial pain syndrome, a condition associated with hypersensitive muscular trigger points * Common myeloid progenitor, otherwise known as CFU-GEMM, the multipotent progenitor cell for the myeloid cell lineage * Comprehensive metabolic panel, a group of 14 blood tests often used in medical diagnosis * Cytidine monophosphate, a DNA nucleotide Military and firearms * Canadian Military Pattern truck, a truck design in World War II * Chief of Military Personnel, the senior Canadian Armed Forces officer responsible for the military's human resource programs * Civilian Marksmanship Program, a U.S. government program that promotes firearms safety training and rifle practice * Compact machine pistol, a class of firearm that encompasses small fully automatic firearms * Corps of Military Police, a fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zipf–Mandelbrot Law
In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it. The probability mass function is given by: :f(k;N,q,s)=\frac where H_ is given by: :H_=\sum_^N \frac which may be thought of as a generalization of a harmonic number. In the formula, k is the rank of the data, and q and s are parameters of the distribution. In the limit as N approaches infinity, this becomes the Hurwitz zeta function \zeta(s,q). For finite N and q=0 the Zipf–Mandelbrot law becomes Zipf's law. For infinite N and q=0 it becomes a Zeta distribution. Applications The distribution of words ranked by their frequency in a random text corpus is approximated by a power-law distribution, known as Zipf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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