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]]
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...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]].
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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
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...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}}
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...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
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''(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