Inversion Sampling
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Inversion Sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u between 0 and 1, interpreted as a probability, and then returns the smallest number x\in\mathbb R such that F(x)\ge u for the cumulative distribution function F of a random variable. For example, imagine that F is the standard normal distribution with mean zero and standard deviation one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution. We are randomly choosing a proportion of the area under the curve and returning the number in the domain such that exactly this proportion of the area occurs t ...
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Nikolai Smirnov (mathematician)
Nikolai Vasilyevich Smirnov (; 17 October 1900 – 2 June 1966) was a Soviet Russian mathematician noted for his work in various fields including probability theory and statistics. Smirnov's principal works in mathematical statistics and probability theory were devoted to the investigation of limit distributions by means of the asymptotic behaviour of multiple integrals as the multiplicity is increased with limit. He was one of the creators of the nonparametric methods in mathematical statistics and of the theory of limit distributions of order statistics. Biography Smirnov was born October 17, 1900, in Moscow into the family of a church clerk who was also employed as a clerk in the office of the Bolshoi Theater. He completed his gymnasium education during the First World War, during which he served in various medical units of the military. After the October Revolution Smirnov joined the ranks of the Red Army. During this time he took an interest in philosophy and philology ...
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Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution Support (measure theory), supported on the real numbers, discrete or "mixed" as well as Continuous variable, continuous, is uniquely identified by a right-continuous Monotonic function, monotone increasing function (a càdlàg function) F \colon \mathbb R \rightarrow [0,1] satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability ...
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Càdlàg
In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space. Two related terms are càglàd, standing for "", the left-right reversal of càdlàg, and càllàl for "" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd. Definition Let (M, d) be a metric space, and let E \subseteq \mathbb. A function f:E \to M is called a càdlàg function if, for every t \in E, * the left limit f(t-) := \lim_f(s) exists; and * the right limit f(t+) := \lim_ ...
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Infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, and if ''b'' is a lower bound of S, then ''b'' is less than or equal to the infimum of S. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; : suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. If the supremum of S exists, it is unique, and if ''b'' is an upper bound of S, then the supremum of S is less than or equal to ''b''. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in anal ...
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Inverse Transformation Method For Exponential Distribution
Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the original number, yields zero * Compositional inverse, a function that "reverses" another function * Inverse element * Inverse function, a function that "reverses" another function **Generalized inverse, a matrix that has some properties of the inverse matrix but not necessarily all of them * Multiplicative inverse (reciprocal), a number which when multiplied by a given number yields the multiplicative identity, 1 ** Inverse matrix of an Invertible matrix Other uses * Invert level, the base interior level of a pipe, trench or tunnel * ''Inverse'' (website), an online magazine * An outdated term for an LGBT person; see Sexual inversion (sexology) See also * Inversion (other) * Inverter (other) * Opposite (disambiguation ...
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Exponential Distribution
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includ ...
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Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ...
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Pseudorandom Number Generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random number generation, random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's ''random seed, seed'' (which may include truly random values). Although sequences that are closer to truly random can be generated using hardware random number generators, ''pseudorandom number generators'' are important in practice for their speed in number generation and their reproducibility. PRNGs are central in applications such as simulations (e.g. for the Monte Carlo method), electronic games (e.g. for procedural generation), and cryptography. Cryptographic applications require the output not to be predictable from earlier outputs, and more cryptographically-secure pseudorandom number generator, elabora ...
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Inverse Transform Sampling Example
Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the original number, yields zero * Compositional inverse, a function that "reverses" another function * Inverse element * Inverse function, a function that "reverses" another function **Generalized inverse, a matrix that has some properties of the inverse matrix but not necessarily all of them * Multiplicative inverse (reciprocal), a number which when multiplied by a given number yields the multiplicative identity, 1 ** Inverse matrix of an Invertible matrix Other uses * Invert level, the base interior level of a pipe, trench or tunnel * ''Inverse'' (website), an online magazine * An outdated term for an LGBT person; see Sexual inversion (sexology) See also * Inversion (other) * Inverter (other) * Opposite (disambiguation ...
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Generalized Inversion Method
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ...
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Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (i.e. ,b/math>) or open (i.e. (a,b)). Therefore, the distribution is often abbreviated U(a,b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is f(x) = \begin \dfrac & ...
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