One of the basic terms in classical statistics and [[Probability theory|probability theory]]. In the axiomatic approach it is defined as any decomposition of t
[[Category:Probability and statistics]]
649 bytes (94 words) - 07:34, 2 November 2014
...iable $X$. The quantiles $K_{1/4}$ and $K_{3/4}$ are called the quartiles, and $K_{0.1},\ldots,K_{0.9}$, the deciles. The values of the quantiles for suit
The quartiles of the normal distribution $\Phi$ are $K_{1/4}=-0.67$ and $K_{3/4}=0.67$.
2 KB (269 words) - 16:40, 5 August 2014
...increasing order, then a point $x_m$ is called a mode if $p_m \ge p_{m-1}$ and $p_m \ge p_{m+1}$.
...with respect to some point $a$, the mode is equal to $a$ and to the median and to the mathematical expectation, if the latter exists.
2 KB (243 words) - 20:49, 14 December 2015
* In probability and statistics: the way to describe probability of random variables taking certain values, see [[Distribution function|Dist
* In differential geometry and topology: [[Distribution of tangent subspaces]].
415 bytes (50 words) - 07:47, 3 May 2012
Any [[probability distribution]] having support on some (linear) manifold of dimension smalle
...to a measure of infinite total mass, which is still being manipulated as a probability distribution. Cf. also [[Improper distribution|Improper distribution]].
1 KB (156 words) - 19:24, 26 October 2014
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...m variables subject to the same absolutely-continuous probability law with probability density $ p( x) $.
4 KB (604 words) - 08:25, 6 June 2020
...sure]] on the real line whose density is zero outside the interval $(0,1)$ and is $(\sqrt{x(1-x)})^{-1}/\pi$ if $0<x<1$. The corresponding distribution fu
...wal theory (cf. [[Arcsine law|Arcsine law]]), and are used in mathematical statistics as special cases of the beta-distribution.
2 KB (226 words) - 12:50, 11 October 2014
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A continuous [[Probability distribution|probability distribution]] with density
3 KB (364 words) - 15:52, 7 June 2020
...ned as the discrete distribution that assigns to every value $ X_{k} $ the probability $ \dfrac{1}{n} $. The empirical distribution function $ F_{n} $ is the step
\dfrac{k}{n}, & \text{if} ~ X_{(k)} < x \leq X_{(k + 1)} ~ \text{and} ~ 1 \leq k \leq n - 1; \\
5 KB (778 words) - 04:08, 22 June 2017
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...f probability mass outside a given interval to within this interval. Let a probability distribution on the line be given by a distribution function $ F $.
4 KB (545 words) - 14:56, 7 June 2020
...},\dots,X_{(n)}$, that is, of the quantity $X_{(k+1)}$ if $n=2k+1$ is odd, and of $[X_{(k)}+X_{(k+1)}]/2$ if $n=2k$ is even.
|valign="top"|{{Ref|L}}|| M. Loève, "Probability theory" , Springer (1977) {{MR|0651017}} {{MR|0651018}} {{ZBL|0359.60001}}
1 KB (227 words) - 22:02, 30 November 2018
...statistical test]] for testing the hypothesis $H_0$ that a one-dimensional probability density is symmetric about zero.
...hypothesis of symmetry $H_0$ be that the probability density $p(x)$ of the probability law of independent random variables $X_1,\ldots,X_n$ is symmetric about zer
1 KB (238 words) - 16:19, 19 April 2014
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A probability distribution on the simplex
2 KB (306 words) - 08:57, 8 April 2023
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...stributions (cf. [[Discrete distribution|Discrete distribution]]) given by probability vectors $ p = ( p _ {1} \dots p _ {n} ) $,
2 KB (316 words) - 22:15, 5 June 2020
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of dispersion for a [[Probability distribution|probability distribution]]. For a continuously-distributed symmetric random variable $
2 KB (234 words) - 08:07, 6 June 2020
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containing with given probability $ \gamma $
7 KB (924 words) - 10:30, 16 July 2021
...sure on $S$ such that $\operatorname P(\Omega)=1$ (a [[Probability measure|probability measure]]).
...terization in different terms (see [[Separable process|Separable process]] and also {{Cite|P}}).
8 KB (1,003 words) - 21:59, 21 November 2018
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...matorstrong consistency" for a sequence of estimators that converges with probability 1 to the value being evaluated.
4 KB (501 words) - 17:46, 4 June 2020
...= 20 : ~/encyclopedia/old_files/data/M064/M.0604610 Moments, method of (in probability theory)
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4 KB (609 words) - 16:25, 6 January 2024
...ferences obtained using it do not depend on the particular null-hypothesis probability distribution of the observable random variables on the basis of which one w
See also [[Non-parametric methods in statistics]]; [[Kolmogorov–Smirnov test]].
2 KB (247 words) - 09:06, 10 April 2023
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A probability distribution with distribution function $ \psi ( a x + b ) $,
2 KB (254 words) - 17:36, 12 January 2021
where $\sigma^2$ is the variance and $\mu$ is the mathematical expectation ($\mu$ must be positive). This expres
[[Category:Probability and statistics]]
500 bytes (77 words) - 19:25, 8 November 2014
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The continuous probability distribution, concentrated on the positive semi-axis $ ( 0, \infty ) $,
4 KB (555 words) - 16:43, 4 June 2020
...lection should be so effected that any group of $n$ elements is drawn with probability $n!\,(N-n)!/N!$. This rule of sampling is realized in determining the winni
...t the sampling rule is defined by a distribution function $F$, so that the probability of obtaining an experimental result comprised in a semi-interval $(a,b]$ is
3 KB (427 words) - 17:10, 30 December 2018
...t a random variable $X$ be subject to the binomial law with parameters $n$ and $\theta$, where $\theta$, $0<\theta<1$, is unknown. The statistic
...nt random variables subject to the same probability law, with a continuous probability density $f(x-\theta)$, $|x|<\infty$, $|\theta|<\infty$. The Pitman estimato
2 KB (258 words) - 09:41, 19 November 2018
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A theorem in the theory of statistical estimation giving a necessary and sufficient condition for a statistic $ T $
4 KB (516 words) - 19:38, 5 June 2020
...diaofmath.org/legacyimages/e/e035/e035060/e0350606.png" /> are independent and identically distributed),
is the density of the standard [[Normal distribution|normal distribution]], and
6 KB (866 words) - 11:09, 12 May 2012
...certainty values of formulas, so that the propagation rule generates upper and lower bounds for the conclusion from those on the premises.
...Q$ can be anything between $0$ and the minimum of the probabilities of $P$ and $Q$.
3 KB (515 words) - 13:25, 18 July 2014
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A continuous probability distribution concentrated on the positive semi-axis $ 0 < x < \infty $
4 KB (556 words) - 18:53, 24 January 2024
[[Probability distribution|probability distribution]] of a random variable $X$ which takes non-negative integer va
...are <math>M</math> "marked" and <math>N-M</math> "unmarked" elements, and if the sampling (without replacement) is performed until the number of "ma
3 KB (447 words) - 14:48, 5 June 2017
...straight line represents the normal distribution function with average 100 and standard deviation 8.
...> N. Arley, K.R. Buch, "Introduction to the theory of probability and statistics" , Wiley (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
1 KB (206 words) - 19:52, 3 May 2023
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A scalar characteristic of the pointedness of the graph of the probability density of a [[Unimodal distribution|unimodal distribution]]. It is used as
4 KB (594 words) - 17:18, 6 January 2024
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The probability distribution of the random variable
5 KB (724 words) - 14:55, 7 June 2020
''(in probability theory)''
...w, the most natural assumption is that $\rho$ and $\theta$ are independent and uniformly distributed in the interval $0\leq\rho\leq1$, $0\leq\theta\leq2\p
2 KB (374 words) - 10:24, 16 March 2023
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''in probability theory and mathematical statistics''
3 KB (414 words) - 16:43, 4 June 2020
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...atural generalization of a [[Binomial distribution|binomial distribution]] and coincides with the latter for $ k = 2 $.
4 KB (522 words) - 13:11, 6 January 2024
$#C+1 = 21 : ~/encyclopedia/old_files/data/S087/S.0807490 Statistics
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3 KB (407 words) - 16:24, 6 January 2024
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A member of the series of order statistics (also called [[variational series]]) based on the results of observations.
12 KB (1,803 words) - 09:08, 10 April 2023
$#C+1 = 39 : ~/encyclopedia/old_files/data/B016/B.0106850 Boltzmann statistics
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5 KB (797 words) - 10:59, 29 May 2020
...e construction of a set of approximate values of the unknown parameters of probability distributions.
in a Euclidean space and let the probability distribution of this vector belong to the parametric family of distribution
8 KB (1,101 words) - 11:37, 10 February 2020
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A discrete probability distribution concentrated on a set of points of the form $ a + nh $,
4 KB (622 words) - 22:15, 5 June 2020
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...bability density (cf. [[Density of a probability distribution|Density of a probability distribution]]) of an observable random vector $ X = ( X _ {1} \dots X _
4 KB (582 words) - 08:05, 6 June 2020
A numerical characteristic of the probability distribution of a random variable. In the most general setting, the mathema
...esgue integral|Lebesgue integral]] with respect to a [[Probability measure|probability measure]] $ {\mathsf P} $
7 KB (945 words) - 11:49, 10 February 2020
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s strategy is a decision rule (decision function), and the pay-off function of player $ I $
3 KB (461 words) - 08:23, 6 June 2020
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A statistic having a fixed probability distribution under some compound hypothesis.
3 KB (460 words) - 08:54, 6 January 2024
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A metric or pseudo-metric on the set of probability distributions, characterizing the "non-similarity" of the random phenomen
7 KB (1,010 words) - 22:12, 5 June 2020
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''for a family of probability distributions $ \{ {P _ \theta } : {\theta \in \Theta } \} $
9 KB (1,282 words) - 08:24, 6 June 2020
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A discrete [[Probability distribution|probability distribution]] of a random variable $ X $
3 KB (407 words) - 08:05, 6 June 2020
...function of a real variable $x$ taking at each $x$ the value equal to the probability of the inequality $X < x$.
...w +\infty} F(x) = 1$. (Sometimes a distribution function is defined as the probability of $X \le x$; it is then right-continuous.)
6 KB (864 words) - 13:51, 12 December 2013
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A numerical characteristic of a [[Probability distribution|probability distribution]]. The moment of order $ k $(
7 KB (1,019 words) - 08:53, 21 January 2024
''in probability theory''
.... be finite) or may not (i.e. be infinite). In modern [[Probability theory|probability theory]] the expectation of a random variable is defined in terms of the [[
7 KB (1,127 words) - 20:19, 27 January 2020
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
<center>Department of Probability Theory, Statistics and
20 KB (3,391 words) - 12:09, 28 October 2023
$#C+1 = 42 : ~/encyclopedia/old_files/data/B017/B.0107210 Bose\ANDEinstein statistics,
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7 KB (998 words) - 06:28, 30 May 2020
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and have exponential density $ \lambda e ^ {- \lambda t } $,
3 KB (480 words) - 08:06, 6 June 2020
$#C+1 = 91 : ~/encyclopedia/old_files/data/D031/D.0301110 Density of a probability distribution,
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6 KB (898 words) - 17:32, 5 June 2020
...ity theory|probability theory]] and [[Mathematical statistics|mathematical statistics]].
...which carries a maximum prize, 500,000 were bought in a certain city, the probability of an inhabitant of that city winning the prize is 500,000 / 10,000,000 = 1
9 KB (1,350 words) - 18:22, 27 May 2012
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A particular [[Probability distribution|probability distribution]] of random variables $ X _ {w} $,
5 KB (690 words) - 06:37, 7 October 2023
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Then the a posteriori probability $ {\mathsf P} (A _ {i} \mid B) $
3 KB (402 words) - 10:33, 29 May 2020
$#C+1 = 32 : ~/encyclopedia/old_files/data/F038/F.0308450 Fermi\ANDDirac statistics,
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5 KB (749 words) - 19:39, 5 June 2020
...rding to which, in certain problems in probability theory and mathematical statistics, an event is considered to be practically impossible if it lies in the regi
..., by assuming that the event $\{|X-a|>3\sigma\}$ is practically impossible and, consequently, the event $\{|X-a|<3\sigma\}$ is practically certain. In thi
1 KB (223 words) - 00:44, 24 December 2018
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''(in statistics)''
9 KB (1,309 words) - 08:04, 6 June 2020
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A vector statistic (cf. [[Statistics|Statistics]]) $ R = ( R _ {1} \dots R _ {n} ) $
6 KB (857 words) - 07:07, 21 June 2022
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The probability distribution defined by the formula
4 KB (602 words) - 22:11, 5 June 2020
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...e they measure the discrepancy between the empirical distribution function and $ F _ {0} $.
5 KB (709 words) - 06:49, 26 March 2023
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
'''Summary.''' K.Jordan, best known for his work on probability theory and finite
11 KB (1,734 words) - 12:02, 1 November 2023
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...stributions can be defined in many ways. Here the definition of A.K. Gupta and T. Varga [[#References|[a4]]] is given.
7 KB (992 words) - 07:26, 28 January 2024
...estimators of unknown parameters. Let $X$ be a random vector for which the probability density $p(x|\theta)$ contains an unknown parameter $\theta \in \Theta$. Th
is called the likelihood equation and its solution $\hat\theta$ is called the maximum-likelihood estimator for $\
1 KB (156 words) - 18:13, 19 November 2016
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A continuous probability distribution on the real line with density
2 KB (302 words) - 04:59, 24 February 2022
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A continuous probability distribution concentrated on $(0,\infty)$ with density
7 KB (873 words) - 07:45, 27 January 2024
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
...inal Probability. Its use in Bayesian Statistics as the Evidence of Models and Bayes Factors'''</center>
10 KB (1,679 words) - 18:34, 29 February 2024
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
...inal Probability. Its use in Bayesian Statistics as the Evidence of Models and Bayes Factors'''</center>
10 KB (1,649 words) - 18:34, 29 February 2024
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A [[Probability distribution|probability distribution]] of a random variable $ X $
4 KB (583 words) - 08:02, 6 June 2020
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and $ S = S ( X ) $
3 KB (442 words) - 06:29, 30 May 2020
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
various original contributions to the probability calculus. He played an
11 KB (1,737 words) - 12:11, 23 November 2023
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is rejected with probability $ \phi _ {n} ( X) $,
6 KB (829 words) - 10:21, 16 July 2021
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...eorem deals with sequences of independent trials, in each one of which the probability of occurrence of some event ( "success" ) is $ p $.
4 KB (524 words) - 10:58, 29 May 2020
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...tions of finite-dimensional random vectors), as well as of random elements and stochastic processes (cf. [[Random element|Random element]]; [[Stochastic p
11 KB (1,515 words) - 08:03, 6 June 2020
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A [[Statistical test|statistical test]] for the hypothesis that the probability density of the random vector under observation is symmetric with respect to
2 KB (290 words) - 08:09, 6 June 2020
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for finite memoryless information sources with known statistics was proposed by D. Huffman in 1952 (see [[#References|[a1]]], [[#References
6 KB (919 words) - 22:11, 5 June 2020
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The name of a family of continuous probability distributions (Pearson distributions) whose densities $ p( x) $
6 KB (779 words) - 07:20, 24 March 2023
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
Jeffreys, a distinguished British geophysicist, advocated and justified the
9 KB (1,388 words) - 13:03, 18 March 2023
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and $ x $
5 KB (653 words) - 20:28, 17 January 2024
...led a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also [[Gamma-function|Gamma-function]].
...ial is extensively used in combinatorics, probability theory, mathematical statistics, etc. Cf. [[Combinatorial analysis|Combinatorial analysis]]; [[Combination]
1 KB (162 words) - 13:41, 14 February 2020
...the choice of $y$ is governed by the significance level, which equals the probability of the event $\{Y>y\}$ in the case that the hypothesis $H$ is true.
...e used, for example, in statistical detection of toxic medical preparates, and also in other special situations where the overriding purpose is to ensure
3 KB (454 words) - 15:45, 4 November 2014
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.... The modern formulation of the Behrens–Fisher problem is due to R. Fisher and is based on the concept of [[sufficient statistic]]s. Let $ X _ {11} \dot
7 KB (881 words) - 10:27, 16 July 2021
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...ucted from a [[Sample|sample]] and the corresponding [[Probability measure|probability measure]]. Let $ X _ {1} \dots X _ {n} , \dots $
9 KB (1,327 words) - 19:37, 5 June 2020
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and is completely determined by the distribution of the original vector. For ex
2 KB (343 words) - 07:40, 14 January 2024
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has a definite probability distribution whose mathematical expectation is a function of $ x $:
10 KB (1,524 words) - 08:10, 6 June 2020
between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample
...\le x\}$ be the distribution function of the random variable $X$. Then the probability distribution for the range is
1 KB (196 words) - 12:46, 14 February 2020
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...multivariate attribute in question, and intended for obtaining scientific and practical inferences. By a multivariate attribute is meant a $ p $-
27 KB (3,850 words) - 14:44, 7 June 2020
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A continuous probability distribution concentrated on the positive semi-axis $ 0 < x < \infty $
5 KB (610 words) - 19:21, 10 April 2024
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
'''Summary.''' Kravchuk was a Professor of Mathematical Statistics in Kyiv,
10 KB (1,403 words) - 16:21, 10 March 2024
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according to which the probability distribution $ {\mathsf P} _ \theta $
3 KB (374 words) - 08:07, 6 June 2020
$#C+1 = 22 : ~/encyclopedia/old_files/data/P074/P.0704940 Probability of large deviations
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4 KB (604 words) - 20:08, 7 January 2024
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...te 1970s. The historical development can be seen from [[#References|[a1]]] and [[#References|[a6]]].
11 KB (1,396 words) - 19:32, 5 June 2020
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be a sampling space and suppose that the hypothesis $ H _ {0} $:
3 KB (429 words) - 22:13, 5 June 2020
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and a [[Riemannian metric|Riemannian metric]] $ g $.
10 KB (1,420 words) - 08:23, 6 June 2020
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A so-called positional parameter, which parametrizes a family of probability distributions of one type. A distribution in $ \mathbf R $
1 KB (203 words) - 08:12, 6 June 2020
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...of discriminatory information [[#References|[a3]]], [[#References|[a4]]], and measures of variation-distance [[#References|[a5]]]. While these indices ha
9 KB (1,271 words) - 10:58, 29 May 2020
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A [[Probability distribution|probability distribution]] of a random variable $ X $
7 KB (971 words) - 17:34, 6 January 2024
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...c. When considering the statistical problem of discrimination, S. Kullback and R.A. Leibler [[#References|[a13]]] introduced a measure of the "distance"
9 KB (1,370 words) - 22:15, 5 June 2020
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
'''STATISTICS IN INSURANCE'''
9 KB (1,305 words) - 09:48, 28 October 2023
...ct, and the natural numerical measure of indefiniteness of an event is its probability.
...e assumption that the methods of encoding the message into an input signal and of decoding the output into a message may vary within wide ranges.
7 KB (993 words) - 17:28, 7 February 2011
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A measure of the linear dependence between one random variable and a certain collection of random variables. More precisely, if $ ( X _ {1}
5 KB (666 words) - 08:02, 6 June 2020
''of stochastic processes and fields''
...odic theory|ergodic theory]] and mixing conditions in [[Probability theory|probability theory]] based on appropriate mixing coefficients measuring the dependence
7 KB (1,086 words) - 06:27, 26 March 2023
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A distance between probability measures, expressed in terms of the [[Hellinger integral|Hellinger integral
3 KB (423 words) - 22:10, 5 June 2020
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and if for some $ n \geq 1 $
5 KB (661 words) - 12:22, 17 January 2021
...re the exposition is based on the concept of a space of elementary events, and where it is stressed that only in this context the notion of a random varia
...quation $P_X(B)=P\{X(\omega)\in B\}$, $B\in\mathcal{B}_1$, is called the [[probability distribution]] of $X$. This measure is uniquely determined by the [[distrib
6 KB (904 words) - 20:43, 1 January 2019
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...rameters of probability laws from results of observations. It was proposed and developed by [[Neyman, Jerzy|J. Neyman]] (see [[#References|[1]]], [[#Refer
8 KB (1,137 words) - 07:29, 24 March 2023
...iance is defined for random variables $X_1$ and $X_2$ with finite variance and is usually denoted by $\cov(X_1, X_2)$. Thus,
If $X_1$ and $X_2$ are independent random variables, then $\cov(X_1, X_2)=0$. The covari
2 KB (232 words) - 10:53, 18 October 2014
$#C+1 = 20 : ~/encyclopedia/old_files/data/R082/R.0802530 Robust statistics
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The probability distribution of a random variable $ X $
5 KB (776 words) - 10:59, 29 May 2020
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and $ \mu _ {3} $
3 KB (461 words) - 07:30, 26 March 2023
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...eed for the introduction of such a concept arose as early as 1955 or 1956, and it was at that time that Le Cam selected the name of "contiguity" , with t
15 KB (2,180 words) - 17:43, 1 July 2020
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The continuous probability distribution, concentrated on $ ( 0 , \infty ) $,
4 KB (482 words) - 22:17, 5 June 2020
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...termination of functions of given arbitrary random variables for which the probability distributions possess given properties.
4 KB (562 words) - 20:33, 16 January 2024
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
'''Summary.''' Rice, American sociologist and statistician,
6 KB (891 words) - 20:38, 22 September 2016
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...bability]] function on subsets of $\mathcal{S}$, assigning a [[Probability|probability]] to every matrix event.
8 KB (1,193 words) - 17:37, 11 November 2023
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The [[Probability distribution|probability distribution]] of the random variable
3 KB (396 words) - 20:47, 22 January 2024
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...e an unforeseen effect on each of the measurements, both in overestimating and in underestimating results.
12 KB (1,805 words) - 08:11, 21 January 2024
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
'''Summary.''' A physician and writer, the creator of John Bull,
10 KB (1,512 words) - 07:24, 25 March 2023
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treatment of outliers in statistics and to the application of probability
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and $ F $
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In probability theory, the quadratic deviation $\sigma_X$ of a random variable $X$ (from i
...c deviation is taken as a measure of the quality of statistical estimators and in this case is referred to as the quadratic error.
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''in probability theory''
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..." , to transmit it over a [[Communication channel|communication channel]], and to transform it into "signals" . The intuitive picture of information rela
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...ondences between experimental data on the one hand and hypotheses on their probability characteristics on the other.
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A general theory for the processing and use of statistical observations. In a broader interpretation of the term, s
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...{ T _ { n } } ( x _ { i } )$. If $f _ { \theta } ( x ) > 0$ for all $x$ and $\theta$, then this is equivalent to minimizing $\sum _ { i = 1 } ^ { n } [
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''for the unknown true value of a scalar parameter of a probability distribution''
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one of the first studied in probability. Thomas Bayes, a nonconformist
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$#C+1 = 58 : ~/encyclopedia/old_files/data/M062/M.0602710 Mathematical statistics
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...riables that can be used in estimating unknown parameters of a theoretical probability distribution. Methods of the theory of statistical estimation form the basi
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...ic probabilities|geometric probabilities]], with the introduction of ideas and methods from the theory of random processes, especially the theory of point
...x polyhedra such that the interiors of the polyhedra do not intersect with probability 1, while $\mathbf R^n$ is equal to the union of their closures), etc.
7 KB (1,051 words) - 09:27, 22 August 2014
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century probability theory. His major probabilistic work, the ''
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...ndom processes. His textbook influenced the development of probability and statistics internationally.
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Davis (2003--07) and at UC Berkeley (1986--89). He received in 2006 the
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...on coefficient between the random variables $Y_1 = X_1 - X^*_{1;3\dots n}$ and $Y_2 = X_2 - X^*_{2;3\dots n}$:
...where $\rho_{ij}$ is the correlation coefficient between $X_i$ and $X_j$, and let $P_{ij}$ be the cofactor of the element $\rho_{ij}$ in the determinant
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...ulation. The fundamental difference between the sampling theory for finite and infinite populations consists in the fact that in the former case the theor
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...f future discounted benefits less premiums. Splitting into payments before and after time $t + d t$ leads to
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administrative and bureaucratic statistics. He was hostile to the use of the
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school of probability
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and let $ mX $
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'''Summary'''. Finnish mathematician and statistician, Lindeberg
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and $ d _ {k} ( n,r ) $
such that, whatever the [[Probability space|probability space]] and the events $ A _ {j} $
8 KB (1,173 words) - 12:05, 9 February 2020
...h $ i \in \{ 1,\ldots,n \} $. For example, with a sample size of $ n = 5 $ and distinct observations $ X_{1},X_{2},X_{3},X_{4},X_{5} $, one might obtain $
...ots,X_{n}) $ (cf. also [[Statistical estimation|statistical estimation]]), and let $ T_{n} \stackrel{\text{df}}{=} T(X_{1},\ldots,X_{n}) $. The object of
9 KB (1,419 words) - 05:52, 15 June 2017
...30 : ~/encyclopedia/old_files/data/C021/C.0201890 Chebyshev inequality in probability theory,
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and $ b _ {i} $
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''Ars Conjectandi'' and Abraham de Moivre's ''De Mensura Sortis''; and on his
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be independent identically distributed random variables with a probability density depending on a parameter $ \theta $(
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A continuous [[probability distribution]] concentrated on $ (0, 1) $
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'''Summary.''' Willcox, demographer and statistician, founded
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A term occurring in the area of statistics called [[Sequential analysis|sequential analysis]]. In sequential procedure
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'''Summary'''. Darmois was the first French mathematician to teach and
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He is an Elected Fellow of the American Statistical Association (2001) and of the International Statistical
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...ached via properties of estimating functions, namely functions of the data and the parameter of interest, rather than estimators derived therefrom. For a
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''in probability theory''
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...bles with the same distribution (cf. [[Random variable|Random variable]]), and let $ S _ {0} = 0 $,
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...al inferences concerning a probability model suggested for the description and explanation of an observed sequence of data.
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(later called Simpson's paradox) and the Yule process all perpetuate the name of a pioneer
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be a continuously-distributed random variable for which the probability density $ p( x) $,
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...= y\mid S_0=0) = C_{\frac{(n+y)}{2}}^{n} p^{(n+y)/2} q^{(n-y)/2}$, if $y$ and $n$ are of the same parity, i.e., if either both are odd or both are even.
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...(see [[Regression|Regression]]). The problem of regression in mathematical statistics is characterized by the fact that there is insufficient information about t
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which states that independent and identically-distributed random variables $ X _{1} \dots X _{n} $
and $ \Psi (t) $
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research in probability, statistics, and insurance mathematics, and to the
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...riori information that an observed random phenomenon can be described by a probability distribution $ P ^ \theta ( d \omega ) $
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administrator and entrepreneur, as well as for her contributions to
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classical gymnasium in this town, dreamed of becoming a physicist, and regarded
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...time series, in non-linear problems of statistics of stochastic processes, and elsewhere (see [[#References|[1]]]–[[#References|[3]]]).
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...ds of randomization, which in turn makes it necessary to use probabilistic and statistical methods.
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French mathematics from 1815 to 1840. His contribution to probability theory is
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...in [[Probability|probability]] and [[Mathematical statistics|mathematical statistics]]. If $ T = [ 0, \infty ) $,
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...of the process. The corresponding integrand has the form of the Lagrangian and has been called the Onsager–Machlup function by physicists.
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'''Summary''' The correspondence between Chuprov and Markov marks the coming
12 KB (1,787 words) - 07:49, 18 August 2023
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theory in areas as diverse as non-parametric inference and the properties
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interest in logical argument, probability and statistics, and his fertility
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sciences, economics, probability and statistics. His early and
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'''Summary.''' As an organizer of statistics, and equally as a
11 KB (1,620 words) - 06:53, 8 October 2023
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statisticians and contributed especially to the use of statistical methods in
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statistics as a highly mathematical subject based on
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theory of probability.
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in Australia, Knibbs contributed significantly to vital statistics.
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and published
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'''Summary''' Bernstein's training in and continuing contacts
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rise of 20th century mathematical statistics, contributing to the
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...hypothesis. The need to measure the efficiency of tests arose in the 1930s and -forties when simple (from the computational point of view) but "inefficie
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'''Summary.''' Egon Pearson continued and developed his father,
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...ributions on the same measurable space) in statistics which is widely used and studied for two reasons:
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such that the family of probability distributions $ \{ { {\mathsf P} _ \theta } : {\theta \in \Theta } \} $
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$#C+1 = 129 : ~/encyclopedia/old_files/data/C026/C.0206560 Correlation (in statistics)
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...obability measure|probability measure]] $\mathsf{P} _ { p }$ describes the probability
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Department of Statistics
22 KB (3,172 words) - 20:20, 12 March 2016
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probability and statistics (randomness) which was rehabilitated in the
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and the equations (1) take the form
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...rs who also make a payment; the recruiter receives part of these receipts, and a cut of future payments as the new recruits go on to recruit others. In re
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'''Summary.''' Founding Director of the Central Bureau of Statistics in
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...a large number of independent or weakly-dependent random variables have a probability distribution close to the [[Normal distribution|normal distribution]].
20 KB (2,992 words) - 07:15, 21 March 2023
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'''Summary.''' A major developer of the foundations of statistics and an important
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'''Summary'''. Boole's name in probability theory and statistical
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horses. A pioneer faithful to his master Lexis, he extended in numerous and
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spread of smallpox, which was prevalent at the time, and argued the advantages of
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general and President of
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and has many interesting properties. For instance, under mild conditions (e.g.,
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'''Summary.''' Arguably the Father of Modern Statistics, Graunt produced
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logic and his compilation of data on alumni of Cambridge University, but
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is a [[Martingale|martingale]] and $ A ( t) $
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...timate the unknown values of these parameters from results of observations
and to test hypotheses concerning their values.
...://www.encyclopediaofmath.org/legacyimages/n/n067/n067230/n0672306.png" />
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
37 KB (5,331 words) - 17:14, 7 February 2011
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and suppose one is testing the hypothesis $ H _ {0} $:
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...oblems of mathematical statistics, when there is a priori information: the probability distribution $ P ( d \omega ) $
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...}$, Stephen T. Buckland$^{a}$, David L. Borchers$^{a}$, Eric Rexstad$^{a}$ and Len Thomas$^{a}$}
9 KB (1,288 words) - 19:03, 28 February 2024
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modelling and data
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is a sequence of random variables and $ Y $
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...Random variable]]) with means $\mu_x=\mathsf{E}(X)$, $\mu_y=\mathsf{E}(Y)$ and finite variances $\sigma^2_x=\text{var}(X)$, $\sigma^2_y=\text{var}(Y)$:
where $\text{cov}(X,Y)$ is the [[Covariance|covariance]] of $X$ and $Y$,
4 KB (683 words) - 07:24, 24 March 2023
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distributions. In statistics, these are standard tools for modelling
17 KB (2,589 words) - 12:44, 28 October 2023
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...des. Overviews are given in Verbeke and Molenberghs (2000) and Molenberghs and Verbeke (2005).
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An inequality in mathematical statistics that establishes a lower bound for the risk corresponding to a quadratic lo
...function $ p(x |\theta) $, where $ x = (x_{1},\ldots,x_{n})^{\intercal} $ and $ \theta \in \Theta \subseteq \mathbb{R} $. Suppose that a statistic $ T =
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...differential]]. More precisely, a continuous stochastic process $X_t$ on a probability space $(\Omega, \mathcal{F}, \Prob)$ with a certain non-decreasing family $
and $ \sigma (t) $ (called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to $ {\
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Aveyron (southern Massif Central), France, and died in Paris. He began his
14 KB (2,139 words) - 18:37, 8 March 2024
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promoted the use of statistical methods in economics and developed early
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...sh literature the Russian name was pronounced typically as "Wasserstein" and the notation $W ( P , Q )$ is common for $\mathbf{l} _ { 1 } ( P , Q )$.
6 KB (948 words) - 16:55, 1 July 2020
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stable non-Gaussian probability distribution.
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interest in statistics than any of the other great economists.
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...iability theory. This is because the events represented by the qualitative and quantitative reliability indices (failure, time before failure, repair time
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trials during the mid-20th century, and conducted important epidemiogic
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He studied conducting and piano at the conservatory
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economics, Cournot was both mathematician and philosopher. He publicised the
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and the sequence of sums
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...e_{\B})=1$ (that is, $\mu$ is a probability measure) in {{Cite|H2|p. 43}} and {{Cite|K|Sect. 17.F}} to $\mu(\bsone_{\B})<\infty$ (that is, $\mu$ is a to
...ses of measurable sets. (The equivalence is [[Measure space#Basic notions and constructions|equality mod 0]]. Sets of the original σ-algebra or its [[
9 KB (1,257 words) - 05:15, 4 September 2013
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Large deviations is concerned with the study of rare events and of
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T R Malthus, English economist, statistician and demographer, was born
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associated with the discovery of the ''t''-distribution and its use, and he
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to make probability its main concept. He formulated and proved the weak law of
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'''Summary.''' Physicist, psychologist and philosopher, Fechner is noted for the
16 KB (2,419 words) - 16:35, 14 August 2023
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and
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to mean values and variability and association between statistical variates,
13 KB (1,954 words) - 07:48, 14 August 2023
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proved the Bienaymé-Chebyshev Inequality some years before Chebyshev, and
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The continuous probability distribution, concentrated on the positive semi-axis $ ( 0, \infty ) $,
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is a set of probability measures on $ ( \Omega, {\mathcal F} ) $.
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are more pertinent than has long been believed, and were the starting point for
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...the frequency of occurrence of a random event tends to become equal to its probability as the number of trials increases (a phenomenon which was probably first no
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analysis, actuarial science, and mathematical statistics. He invented the
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vital statistics and
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biology, and
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integral part of statistics and his theory of errors remained a major focus of
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...or less non-precise. This uncertainty is different from measurement errors and has to be described formally in order to obtain realistic results by mathem
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introduced several original ideas in probability and statistics, notably the
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In contrast to classical information, described by means of a probability distribution on the space of signals $ X $,
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socialists and slums, Bowley, a pioneer in the development of
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was not her lamp but her pen, combined with administrative skill and
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One of the fundamental parts of mathematical statistics, dedicated to the estimation using random observations of various character
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...a binomial random variable $T$ with distribution $B(n,p)$, with mean $np$ and variance $npq (q=1-p)$, the normal approximation is given , for integers $0
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of mathematics, which encompassed path-breaking work in [[probability theory]]. The
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was a notable chemist and had a significant role in developing
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statistical sampling and approaches to quality improvement, Deming led
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Swiss official statistics, Franscini was both politician and scientist.
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including normality tests, stochastic ratios and time series analysis,
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Many national statistical agencies, survey organizations, and
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'''Summary.''' Condorcet applied mathematics in precocious and philosophically
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'''Summary.''' A father figure in Indian Statistics,
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...al distribution if $X := \Gamma X$ for all $\Gamma \in \mathcal{O} ( p )$; and
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field, and Lenin used his data. As an émigré professor in Prague, he
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astronomy and meteorology,
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'''Summary''' Pascal introduced the concept of mathematical expectation and
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...sed estimator]]). If the random errors in the observations are independent and follow a normal distribution law, the method yields estimators with minimum
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non-negative integers, are intrinsically heteroskedastic, right skewed, and have a variance
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...gation of the general laws governing the structure, composition, dynamics, and evolution of stellar systems.
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of Statistics,
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...resently (1996), the best sources for information are [[#References|[a1]]] and [[#References|[a2]]].
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Institute for Statistics and Econometrics, Georg-August-Universität
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chiefly remembered today for his work in algebra and logic. He also made
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...ypotheses (determined by the Neyman–Pearson lemma) for a fixed sample size and the same error probabilities.
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<center>''' The Significance Test Controversy and the Bayesian Alternative'''</center>
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and $ b( t, X) $
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Statistics. This problem became especially important in the middle
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In mathematical statistics and probability theory one uses the Hermite polynomials corresponding to the weight functio
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<ref> Université Paris Dauphine, 75785 Paris cedex 16, France, and CREST, INSEE. Email: xian@ceremade.dauphine.fr </ref>
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<center> Simo Puntanen <ref> Department of Mathematics and Statistics,
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...od; queue length; steady state probability; stochastic process; transition probability; waiting line.
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...ackground of an action of secondary random factors and auxiliary phenomena and, as a consequence, the aim to detect theoretically-predictable relations wi
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of one of the few treatises of probability theory in the 19th century. He
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and let
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...$ needs $3$ times as much observations to perform equally well as test $1$ and hence test $1$ is $3$ times as efficient as test $2$ (cf. also [[Efficient
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Professor, Department of Statistics
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A branch of mathematical statistics devoted to statistical inferences on the basis of observations represented
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...which are directed towards the agreement between the data on the one hand and empirically verifiable components of a model on the other.
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on a probability space $ ( \Omega , {\mathcal A} , {\mathsf P}) $
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Jill M. Montaquila and Graham Kalton
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'''Summary.''' The data of Pastor Caspar Neumann on births and funerals in Breslau
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probability calculus, but it is above all for his role as teacher, as publicist,
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..._{iq},y_{i}\}$ represent the $i$th observation on a set of $q+1$ variables and suppose that we wish to fit a
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...irst developed by John Nelder and R.W.M. Wedderburn in 1972, the algorithm and
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branching process with immigration, and of statistical inference for a random process.
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is unknown, and if it is necessary to estimate the value of a given function $ \phi (X) $
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The basic paired comparison model was presented by R.A. Bradley and M.E. Terry in [[#References|[a1]]] (cf. also [[Bradley–Terry model|Bradle
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[[Correlation|Correlation]] between variables in multivariate statistics (cf. also [[Multi-dimensional statistical analysis|Multi-dimensional statis
18 KB (2,717 words) - 20:14, 8 December 2023
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political circumstances, Christiaan Huygens turned to science and mathematics.
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, Eds. P. Armitage and T. Colton, and published by kind permission
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...main, part of it was devoted to the principles of human use of a language and the following was the main thesis of the author [[#References|[a25]]], pp.
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support himself by tutoring and he graduated in 1840. In 1843, after mental
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and obtained numerous results which are both remarkable and
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middle class and aristocracy. He was a student at the prestigious
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of queueing theory for application to telephone traffic and was a precursor
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employed in the hard copy edition of Statisticians of the Centuries and elsewhere is actually that of the politician Louis Legendre (1752--1797).
11 KB (1,629 words) - 11:46, 23 November 2023
$#C+1 = 222 : ~/encyclopedia/old_files/data/P074/P.0704970 Probability theory
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42 KB (6,282 words) - 08:07, 6 June 2020
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the statistics of his day from a modest collection of useful ad hoc
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statistics.
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'''Incomplete Data in Clinical and Epidemiological Studies'''
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...odynamics, electromagnetism, quantum mechanics, plasma physics, chemistry, and so forth – also apply to distant cosmic phenomena.
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medical doctor and capable mathematician, pioneered
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and $B$~is a random partition of the index set~$\Nat$.
10 KB (1,615 words) - 10:21, 10 March 2024
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...game show. The host, a certain Mr. Monty Hall, shows you three large doors and tells you that behind one of the doors there is a car while behind the othe
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...-dimensional statistical analysis]], serving for a quantitative expression and processing of the available information in accordance with the criterion fo
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and
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physical and engineering sciences.
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<center>'''Nonparametric regression using kernel and spline methods'''</center>
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games of chance and of annuities on lives for his clients whom he used to meet
19 KB (3,099 words) - 16:34, 14 August 2023
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lines and correlation between variables. He helped in founding the
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His methods were closely related to an epistemology of physics and
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the (Jesuit) ''Collegium Ragusinum'' in Ragusa and at the
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professor of economics. Otherwise, his "pillars and arches", as
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In 1903 he treated a stochastic model for an insurance business and
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and a short note such as this cannot even begin to do
23 KB (3,684 words) - 18:32, 17 April 2024
...gates. Since $M(\alpha)$ depends only on $f$, it is also denoted by $M(f)$ and called the Mahler measure of $f$. Jensen's formula (cf. also [[Jensen formu
and this observation permits one to generalize Mahler's measure to polynomials
7 KB (1,029 words) - 07:50, 27 March 2018
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that is also reflexive and transitive;
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...[[#References|[a4]]] and subsequently proved to be one of the most useful and versatile statistical models, in particular with regards to applications in
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and $ f \in L _ {2} $;
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century, the leading figure in all of stochastics, and he has
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''in mathematical statistics''
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Department of Statistics
29 KB (4,005 words) - 20:28, 13 March 2024
consisting of zeros and ones. Any stochastic matrix $P$ can be
indecomposable and if the class of positive states of the Markov chain
6 KB (982 words) - 09:11, 1 October 2023
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to be realistic. A general, and very flexible, class of parametric models for continuous longitudinal data
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...said to learn from experience $E$ with respect to some class of tasks $T$ and performance measure $\pi$, if its performance at tasks $t \in T$, as measur
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individual, and $x_i=(x_{i1},\ldots,x_{iv},\ldots,x_{ip})'$ be the associated
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<center>'''Random Coefficient Models\footnote {N. T. Longford, SNTL and Departament d'Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Far
23 KB (3,576 words) - 22:10, 12 March 2016
...technology and natural science, the accumulation of quantitative relations and spatial forms studied in mathematics is continuously expanding; so this gen
...with very simple aspects of economic life. The first problems of mechanics and physics were already in this collection of fundamental mathematical ideas.
22 KB (3,256 words) - 21:54, 30 March 2012
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form of 1/0, with 1 generally indicating a success and 0 a failure.
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...stical thermodynamics. It appears also in other interpretations in lattice statistics: defining occupation variables $\rho _ { i } = ( 1 - S _ { i } ) / 2$, wher
16 KB (2,407 words) - 19:47, 17 February 2024
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phenomena. He was a fine scientist and made fundamental contributions to
12 KB (1,866 words) - 14:20, 10 March 2024
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''London School of Economics and Political Science'' </center>
15 KB (2,396 words) - 20:19, 12 March 2016
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Department of Computing and Mathematics, University of Derby,
13 KB (1,912 words) - 18:57, 7 March 2024
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...s back to the problem of isolating a signal from a "mixture" of a signal and a random noise. An important case of this is the problem of optimal filteri
15 KB (2,302 words) - 08:03, 21 January 2024
...been made in the theory of controlled jump (or stepwise) Markov processes
and controlled diffusion processes when the complete evolution of the process i
...//www.encyclopediaofmath.org/legacyimages/c/c026/c026010/c02601030.png" />
and control action <img align="absmiddle" border="0" src="https://www.encyclope
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...the algorithm starts with an initial value $\theta ^ { ( 0 ) } \in \Theta$ and iterates until convergence the two following steps for $t = 0,1 , \ldots$:
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...= ( t _ { 1 } , \dots , t _ { N } ) : t _ { i } \geq 0 \}$ with zero mean and covariance function
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poverty, and with policies and insurance against it, was expressed in
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...presence of numerous eddies of varying sizes and therefore change in space and time in a very irregular fashion).
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and "Father
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Demographers study population dynamics: changes in population size and
20 KB (2,970 words) - 20:18, 12 March 2016
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...y has been constructed for various spaces, primarily Euclidean, projective and homogeneous spaces.
28 KB (4,292 words) - 08:45, 1 July 2022
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...discrete data, then called the log-linear model; see [[#References|[a6]]], and [[#References|[a10]]].
56 KB (8,477 words) - 17:45, 1 July 2020
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...e a wide class of problems in mathematical physics and certain problems in probability theory.
22 KB (3,445 words) - 09:41, 15 August 2023
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
Vector autoregressive (VAR) processes are popular in economics and
12 KB (2,037 words) - 08:10, 10 March 2024
...and is frequently used as an example or test case in various mathematical and physical studies of self-similarity. Although it is geometrically more comp
The Sierpiński pre-gaskets (left) and the Sierpiński gasket (right)
26 KB (3,837 words) - 19:21, 17 March 2024
...degrees of freedom. Quantum field theory unifies the description of fields and particles, which, in classical physics, appear as two distinct entities.
...d, as this is the only field having a clear content, both in the classical and the quantum case.
35 KB (5,270 words) - 23:26, 6 December 2016
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and a given "terminal" criterion
32 KB (4,621 words) - 08:04, 6 June 2020
...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic
'''Summary.''' William Playfair, engineer, political economist and
15 KB (2,346 words) - 20:38, 22 September 2016
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...nt problem in the real domain is due to T.J. Stieltjes (1894). He proposed and solved the following problem in connection with the study of continued frac
33 KB (4,743 words) - 01:58, 17 June 2022
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...ous problems employed in many branches of mathematics, mechanics, physics, and technology. In this article the principal ideas of perturbation theory are
35 KB (5,181 words) - 07:24, 8 June 2020
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...system is usually characterized by its position (configuration, location) and the rate of change of this position, while a law of motion describes the ra
27 KB (4,058 words) - 19:36, 5 June 2020