''in probability theory''
...]] or the [[Density of a probability distribution|density of a probability distribution]].
337 bytes (43 words) - 17:03, 7 February 2011
[[Category:Distribution theory]]
...respectively, is such a mixture. The distribution function of a continuous distribution is a continuous function. The absolutely-continuous distributions occupy a
6 KB (843 words) - 11:58, 20 October 2012
The probability distribution concentrated on $(0,\infty)$ with density
...mber $\mu>0$ are parameters. The [[characteristic function]] of the Erlang distribution has the form
2 KB (279 words) - 19:37, 4 December 2016
[[Category:Distribution theory]]
...al $(0,1)$ and is $(\sqrt{x(1-x)})^{-1}/\pi$ if $0<x<1$. The corresponding distribution function is equal to $(2/\pi)\arcsin\sqrt x$ for $0\leq x\leq1$.
2 KB (226 words) - 12:50, 11 October 2014
$#C+1 = 12 : ~/encyclopedia/old_files/data/G044/G.0404230 Geometric distribution
[[Category:Distribution theory]]
1 KB (202 words) - 19:41, 5 June 2020
$#C+1 = 10 : ~/encyclopedia/old_files/data/R077/R.0707730 Rayleigh distribution
A continuous [[Probability distribution|probability distribution]] with density
3 KB (338 words) - 12:55, 17 March 2023
$#C+1 = 9 : ~/encyclopedia/old_files/data/L060/L.0600780 Logistic distribution
A probability distribution with distribution function $ \psi ( a x + b ) $,
2 KB (254 words) - 17:36, 12 January 2021
$#C+1 = 24 : ~/encyclopedia/old_files/data/T094/T.0904360 Truncated distribution
...in 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
...of the hypothetical distribution. The straight line represents the normal distribution function with average 100 and standard deviation 8.
...>[1]</TD> <TD valign="top"> N. Arley, K.R. Buch, "Introduction to the theory of probability and statistics" , Wiley (1950)</TD></TR><TR><TD valign="top
1 KB (206 words) - 19:52, 3 May 2023
A continuous probability distribution with density
...s a discontinuity at $x=\beta$. The characteristic function of the Laplace distribution with parameters $\alpha$ and $\beta$ is
2 KB (326 words) - 07:53, 20 August 2014
$#C+1 = 16 : ~/encyclopedia/old_files/data/D033/D.0303060 Discrete distribution
[[Category:Distribution theory]]
2 KB (315 words) - 19:35, 5 June 2020
$#C+1 = 17 : ~/encyclopedia/old_files/data/M063/M.0603130 Maxwell distribution
The [[Probability distribution|probability distribution]] with probability density
3 KB (351 words) - 11:10, 9 April 2023
$#C+1 = 20 : ~/encyclopedia/old_files/data/M062/M.0602340 Marginal distribution
...iven distribution. Thus the marginal distribution is the projection of the distribution of the random vector $ X= ( X _ {1} \dots X _ {n} ) $
2 KB (343 words) - 07:40, 14 January 2024
$#C+1 = 36 : ~/encyclopedia/old_files/data/N066/N.0606180 Negative binomial distribution
A [[Probability distribution|probability distribution]] of a random variable $ X $
4 KB (583 words) - 08:02, 6 June 2020
...distribution determined by the simple hypothesis is called the hypothesis distribution. E.g., if one observes a random variable $X$, then the statement "X is sub
.../TD> <TD valign="top"> A.M. Mood, F.A. Graybill, "Introduction to the theory of statistics" , McGraw-Hill (1963) pp. §12.2</TD></TR></table>
675 bytes (94 words) - 09:30, 27 April 2014
A [[Probability distribution|probability distribution]] on $\mathbf R^n$ concentrated on a set of [[Lebesgue measure|Lebesgue mea
...uivalent to the following: A distribution is singular if the corresponding distribution function is continuous and its set of growth points has Lebesgue measure ze
2 KB (359 words) - 18:13, 3 August 2014
$#C+1 = 29 : ~/encyclopedia/old_files/data/P071/P.0701750 Pascal distribution
A discrete [[Probability distribution|probability distribution]] of a random variable $ X $
3 KB (407 words) - 08:05, 6 June 2020
...18 : ~/encyclopedia/old_files/data/C022/C.0202100 \BQT Chi\AAhsquared\EQT distribution,
distribution''
4 KB (555 words) - 16:43, 4 June 2020
$#C+1 = 33 : ~/encyclopedia/old_files/data/C020/C.0200850 Cauchy distribution
[[Category:Distribution theory]]
4 KB (498 words) - 15:35, 4 June 2020
...eory]] characterising those [[additive function]]s that possess a limiting distribution.
==Limiting distribution==
2 KB (230 words) - 12:04, 23 November 2023
...o be contrasted with its unconditional or [[A priori distribution|a priori distribution]].
...the conditional density of $X$ when $\Theta=\theta$; then the a posteriori distribution of $\Theta$ for a given $X=x$, according to the [[Bayes formula|Bayes formu
2 KB (248 words) - 21:34, 1 January 2019
...of]]; [[Least squares, method of|Least squares, method of]]). Thus, in the theory of (observational) errors, developed by Gauss for problems in astronomy and
...ribution appears as the limit form of the [[Binomial distribution|binomial distribution]] (with $p=1/2$) had been discovered by A. de Moivre.
2 KB (312 words) - 16:10, 19 August 2014
$#C+1 = 25 : ~/encyclopedia/old_files/data/W098/W.0908040 Wishart distribution
...ution]]. Let the results of observations have a $ p $-dimensional normal distribution $ N( \mu , \Sigma ) $
3 KB (370 words) - 01:20, 19 January 2022
...discrete distributions: If the values $x_k$ of a random variable $X$ with distribution $p_k = \mathsf{P}(X = x_k)$ are arranged in increasing order, then a point
...and mathematical statistics are the unimodal distributions (cf. [[Unimodal distribution]]). Along with the [[mathematical expectation]] and the [[Median (in statis
2 KB (243 words) - 20:49, 14 December 2015
''$F$-distribution, Fisher–Snedecor distribution, Snedecor distribution''
A continuous probability distribution concentrated on $(0,\infty)$ with density
7 KB (873 words) - 07:45, 27 January 2024
The density of a [[Conditional distribution|conditional distribution]]. Let $ ( \Omega , {\mathcal A} , {\mathsf P} ) $
be the conditional distribution of $ X $
2 KB (296 words) - 17:46, 4 June 2020
...value|Exceptional value]]; [[Value-distribution theory|Value-distribution theory]].
...p">[3]</TD> <TD valign="top"> A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)</TD></TR></table>
3 KB (516 words) - 17:32, 5 June 2020
$#C+1 = 40 : ~/encyclopedia/old_files/data/M065/M.0605330 Multinomial distribution,
''polynomial distribution''
4 KB (522 words) - 13:11, 6 January 2024
$#C+1 = 16 : ~/encyclopedia/old_files/data/D032/D.0302840 Dirichlet distribution
A probability distribution on the simplex
2 KB (306 words) - 08:57, 8 April 2023
...de with limiting forms of the [[Hypergeometric distribution|hypergeometric distribution]]. Pearson curves are classified in accordance with the character of the ro
...Pearson curves consists of 12 types plus the [[Normal distribution|normal distribution]]. Many very important distributions in mathematical statistics may be obta
6 KB (779 words) - 07:20, 24 March 2023
Khinchin's theorem on the factorization of distributions: Any probability distribution $P$ admits (in the convolution semi-group of probability distributions) a f
...ite or countable set of indecomposable distributions (cf. [[Indecomposable distribution]]). The factorization (1) is not unique, in general.
2 KB (326 words) - 16:26, 9 April 2016
$#C+1 = 15 : ~/encyclopedia/old_files/data/J054/J.0504260 Joint distribution
A general term referring to the distribution of several random variables defined on the same probability space. Let $
2 KB (319 words) - 22:14, 5 June 2020
$#C+1 = 34 : ~/encyclopedia/old_files/data/U095/U.0905330 Unimodal distribution,
''single-peak distribution''
5 KB (663 words) - 07:36, 10 April 2023
distribution''
A continuous probability distribution concentrated on the positive semi-axis $ 0 < x < \infty $
4 KB (556 words) - 18:53, 24 January 2024
$#C+1 = 58 : ~/encyclopedia/old_files/data/C024/C.0204480 Conditional distribution
...or each fixed elementary event is a [[Probability distribution|probability distribution]] and for each fixed Borel set is a [[Conditional probability|conditional p
4 KB (599 words) - 17:46, 4 June 2020
A continuous probability distribution concentrated on the positive semi-axis $ 0 < x < \infty $
The corresponding distribution function for $ x \leq 0 $
5 KB (610 words) - 19:21, 10 April 2024
''domain of attraction of a stable distribution''
[[Category:Distribution theory]]
3 KB (488 words) - 20:56, 1 January 2019
$#C+1 = 38 : ~/encyclopedia/old_files/data/L057/L.0507640 Lattice distribution
A discrete probability distribution concentrated on a set of points of the form $ a + nh $,
4 KB (622 words) - 22:15, 5 June 2020
[[Category:Distribution theory]]
...tion]], the uniform distribution appears in probability theory as an exact distribution in some problems and as a limit in others.
6 KB (911 words) - 12:00, 22 December 2019
...variable $X$ is being tested, using a test based on a statistic $T(X)$ the distribution function of which — provided $H_0$ is true — is $G(t)$. If the critical
...><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)</TD></TR></table>
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''sample distribution''
...s to every value $ X_{k} $ the probability $ \dfrac{1}{n} $. The empirical distribution function $ F_{n} $ is the step-function with steps of multiples of $ \dfrac
5 KB (778 words) - 04:08, 22 June 2017
be a random variable having a continuous and strictly increasing distribution function $ F $.
has a uniform distribution on the interval $ [ 0 , 1 ] $,
4 KB (562 words) - 20:33, 16 January 2024
...tatistical problem connected with the study of other parameters of a given distribution. More precisely, for a realization of a random variable $ X $,
...to make a statistical inference not depending on these parameters. In the theory of statistical hypothesis testing one often achieves this by narrowing the
2 KB (341 words) - 08:03, 6 June 2020
...al distribution]]. It is used as a certain measure of the deviation of the distribution in question from the normal one. The excess $ \gamma _ {2} $
is the second Pearson coefficient (cf. [[Pearson distribution|Pearson distribution]]), and $ \mu _ {2} $
4 KB (594 words) - 17:18, 6 January 2024
distribution''
The [[Probability distribution|probability distribution]] of the random variable
3 KB (396 words) - 20:47, 22 January 2024
''in probability theory and mathematical statistics''
Theorems that establish a connection between the type of the distribution of random variables or random vectors and certain general properties of fun
3 KB (414 words) - 16:43, 4 June 2020
$#C+1 = 36 : ~/encyclopedia/old_files/data/H048/H.0408430 Hypergeometric distribution
The probability distribution defined by the formula
4 KB (602 words) - 22:11, 5 June 2020
...h that the distributions of $X_2$ and $BX_1+A$ coincide. The corresponding distribution functions are then connected by the relation
Thus the set of distribution functions decomposes into mutually disjoint types. For example, all normal
4 KB (649 words) - 14:15, 17 April 2014
...yclopedia/old_files/data/M064/M.0604610 Moments, method of (in probability theory)
A method for determining a [[Probability distribution|probability distribution]] by its moments (cf. [[Moment|Moment]]). Theoretically the method of momen
4 KB (609 words) - 16:25, 6 January 2024
$#C+1 = 42 : ~/encyclopedia/old_files/data/I052/I.0502550 Involutive distribution
A $ p $-dimensional distribution (or a differential system of dimension $ p $)
3 KB (378 words) - 10:11, 21 March 2022
$#C+1 = 35 : ~/encyclopedia/old_files/data/W097/W.0907370 Weibull distribution
A particular [[Probability distribution|probability distribution]] of random variables $ X _ {w} $,
5 KB (690 words) - 06:37, 7 October 2023
...istributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). If the distributions of $ S _ {n} $
converge to a limit distribution, $ k _ {n} \rightarrow \infty $,
2 KB (324 words) - 18:48, 5 April 2020
...ocess with independent increments is called homogeneous if the probability distribution of $ X ( \alpha + h ) - X ( \alpha ) $,
...can only have discontinuities of the first kind, with probability one. The distribution of the values of such a process for any $ t $
3 KB (473 words) - 08:23, 6 June 2020
$#C+1 = 51 : ~/encyclopedia/old_files/data/P073/P.0703280 Poisson distribution
[[Category:Distribution theory]]
7 KB (971 words) - 17:34, 6 January 2024
$#C+1 = 71 : ~/encyclopedia/old_files/data/P073/P.0703540 P\Aeolya distribution
The probability distribution of a random variable $ X _ {n} $
6 KB (838 words) - 08:08, 6 June 2020
$#C+1 = 18 : ~/encyclopedia/old_files/data/E036/E.0306900 Exponential distribution
[[Category:Distribution theory]]
3 KB (485 words) - 19:23, 10 April 2024
...ble if it lies in the region of values of the [[Normal distribution|normal distribution]] of a random variable at a distance from its [[Mathematical expectation|ma
where $\Phi(\cdot)$ is the distribution function of the standard normal law; whence, in particular, for $k=3$ it fo
1 KB (223 words) - 00:44, 24 December 2018
One of the basic concepts in [[Probability theory|probability theory]] and [[Mathematical statistics|mathematical statistics]]. In the modern ap
..., proved to be too general in the course of the further development of the theory and was replaced by more restrictive ones in order to exclude some "patholo
8 KB (1,003 words) - 21:59, 21 November 2018
$#C+1 = 50 : ~/encyclopedia/old_files/data/S087/S.0807110 Stable distribution
[[Category:Distribution theory]]
6 KB (799 words) - 18:06, 14 October 2023
...R><TD valign="top">[a3]</TD> <TD valign="top"> A.J. Lee, "U-statistics. Theory and practice" , ''Statistics textbooks and monographs'' , '''110''' , M. De
4 KB (507 words) - 22:10, 5 June 2020
...the parameters determining the equilibrium (cf. [[Gibbs distribution|Gibbs distribution]]; [[Gibbs statistical aggregate|Gibbs statistical aggregate]]).
...</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Ya.G. Sinai, "Theory of phase transitions" , Pergamon (1982) (Translated from Russian)</TD></T
1 KB (187 words) - 17:28, 7 February 2011
is the distribution function of the random variable. Another variable which is also occasionall
.... Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , '''3. Design and analysis''' , Griffin (1969)</TD></TR></table>
1 KB (158 words) - 19:42, 5 June 2020
...oloid. The best known result is Linnik's theorem on the asymptotic uniform distribution of integral points over the surfaces of spheres of increasing radius (see [
.... Malyshev, "A new version of the ergodic method of Yu.V. Linnik in number theory" ''J. Soviet Math.'' , '''11''' (1978) pp. 346–352 ''Zap. Nauchn. Sem. Le
3 KB (340 words) - 17:57, 19 October 2014
...r consideration are much larger than the time needed for the local Maxwell distribution to be established, the quantity $(-h(t,\mathbf r))$ may be identified with
The theory was presented by L. Boltzmann in 1872.
2 KB (326 words) - 17:14, 30 December 2018
...ndence and have the same exponential distribution. An elementary flow with distribution
is a particular case of a renewal process (cf. [[Renewal theory|Renewal theory]]). To an elementary flow is related the [[Poisson process|Poisson process]
2 KB (312 words) - 19:37, 5 June 2020
...meromorphic functions (see [[Value-distribution theory|Value-distribution theory]]). Let $ f ( z) $
For this reason, the theory of value distribution of meromorphic functions concerns itself with questions about the asymptoti
6 KB (966 words) - 08:02, 6 June 2020
have a given continuous distribution function $ F{ ( x) } $,
is the empirical distribution function constructed with respect to the sample $ X _ {1} \dots X _ {n} $
6 KB (730 words) - 12:16, 8 June 2020
[[Category:Distribution theory]]
Every distribution function $F(x)$ has the following properties:
6 KB (864 words) - 13:51, 12 December 2013
For discrete distributions (cf. [[Discrete distribution|Discrete distribution]]) given by probability vectors $ p = ( p _ {1} \dots p _ {n} ) $,
cf. [[Density of a probability distribution|Density of a probability distribution]]).
2 KB (316 words) - 22:15, 5 June 2020
...hypotheses on central problems in [[Analytic number theory|analytic number theory]], advanced by I.M. Vinogradov [[#References|[1]]], [[#References|[2]]] at
==Hypotheses on the distribution of power residues and non-residues.==
3 KB (433 words) - 09:08, 2 January 2021
$#C+1 = 49 : ~/encyclopedia/old_files/data/B016/B.0106420 Binomial distribution,
''Bernoulli distribution''
5 KB (776 words) - 10:59, 29 May 2020
...ernoulli trials are one of the principal schemes considered in probability theory.
has a [[Binomial distribution|binomial distribution]]. As $ n \rightarrow \infty $,
4 KB (636 words) - 13:12, 6 February 2020
...f its counting function $N(r,a,f)$, which characterizes the density of the distribution of $a$-points of $f(z)$, and the proximity function $m(r,a,f)$, which chara
...n="top">[5]</TD> <TD valign="top"> P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" ''Acta Math.'' , '''
3 KB (566 words) - 16:17, 1 April 2017
In the limit theorems of probability theory, a fundamental problem is to determine the limiting behaviour, as $ n \ri
...istributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). Suppose that the sequence of series $ \xi _ {nk} $
5 KB (647 words) - 08:13, 6 June 2020
$#C+1 = 55 : ~/encyclopedia/old_files/data/L057/L.0507760 Least\AAhfavourable distribution
An [[A priori distribution|a priori distribution]] maximizing the risk function in a statistical problem of decision making.
5 KB (695 words) - 22:16, 5 June 2020
A numerical characteristic of a [[Probability distribution|probability distribution]]. The moment of order $ k $(
is the [[Distribution function|distribution function]] of the random variable $ X $,
7 KB (1,019 words) - 08:53, 21 January 2024
...(cf. also [[Density of a probability distribution|Density of a probability distribution]]) has the form
...functions of a random matrix with a matrix variate elliptically contoured distribution also have elliptically contoured distributions. That is, if $X \sim E _ { p
7 KB (992 words) - 07:26, 28 January 2024
...mal distribution|normal distribution]], the [[Poisson distribution|Poisson distribution]] and their compositions (cf. [[Lévy–Cramér theorem|Lévy–Cramér the
...to $\mathfrak L$. This condition is not sufficient, but it is known that a distribution of $\mathfrak L$ belongs to $I_0$ if
4 KB (636 words) - 17:54, 13 November 2014
$#C+1 = 46 : ~/encyclopedia/old_files/data/B016/B.0106810 Boltzmann distribution
The statistical equilibrium distribution function $ f ( \mathbf p , \mathbf r ) $
5 KB (762 words) - 10:59, 29 May 2020
having a [[Poisson distribution|Poisson distribution]]. In the homogeneous Poisson process
One of the properties of a Poisson process is that the conditional distribution of the jump points $ 0 < \tau _ {1} < \dots < \tau _ {n} < t $
3 KB (480 words) - 08:06, 6 June 2020
converges in distribution to $ f ( W ) $,
is a Wiener random function. Thus, the limiting distribution for the $ f ( Y _ {n} ) $
3 KB (432 words) - 19:45, 16 January 2024
...sen independently of the others and in accordance with a given probability distribution. Sometimes a random coding is defined in such a way that every realization
...><TD valign="top">[3]</TD> <TD valign="top"> R. Gallagher, "Information theory and reliable communication" , Wiley (1968)</TD></TR></table>
2 KB (258 words) - 17:04, 7 February 2011
have a given continuous distribution function $ F (x) $.
is the [[Empirical distribution|empirical distribution]] function constructed from the sample $ X _{1} \dots X _{n} $
4 KB (572 words) - 11:08, 26 March 2023
...Goodness-of-fit test|Goodness-of-fit test]]) one wants to test whether the distribution function of a [[Random variable|random variable]] $ X $
...e simplest case this set consists of one completely specified (continuous) distribution function $ F _ {0} $,
5 KB (709 words) - 06:49, 26 March 2023
...s distribution in Gauss' theory of errors (cf. [[Errors, theory of|Errors, theory of]]). The densities
2 KB (315 words) - 19:41, 5 June 2020
...sion of the [[Central limit theorem|Central Limit Theorem]] of probability theory: If $ S_{n} $ denotes the number of “successes” in $ n $ [[Bernoulli tr
is the cumulative distribution function of the standard normal law.
5 KB (692 words) - 19:33, 7 July 2016
...in an arcsine distribution or a generalized [[Arcsine distribution|arcsine distribution]]. The following feature of a Brownian motion $ \{ {\xi _ {t} } : {t \geq
will then have the arcsine distribution:
4 KB (576 words) - 18:48, 5 April 2020
that is, the [[Joint distribution|joint distribution]] of $ X _ {1} \dots X _ {k} $
To test the hypothesis of no relationship, the sampling distribution of $ r _ {1 \cdot ( 2 \dots k) } $
5 KB (666 words) - 08:02, 6 June 2020
...p=0.25$, has the name inter-quartile distance, and in the case of a normal distribution it is equal to $1.349\sigma$ (where $\sigma$ is the natural measure of disp
...ign="top">[1]</TD> <TD valign="top"> G.U. Yale, "An introduction to the theory of statistics" , Griffin (1916)</TD></TR></table>
1 KB (190 words) - 21:50, 9 November 2014
...oper distributions is a normal distribution, then each of them is a normal distribution; and 2) if $\phi_1(t)$ and $\phi_2(t)$ are characteristic functions and if
...mplies closeness of the distribution of each of the summands to the normal distribution; qualitative estimates of the stability are known.
4 KB (647 words) - 19:21, 24 March 2023
...e, instead of the term "non-parametric test" one speaks frequently of a "distribution-free test" . The [[Kolmogorov test]] is a classic example of a non-parametr
....Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian)</TD></TR>
2 KB (247 words) - 09:06, 10 April 2023
...(for a classical system — towards the local [[Maxwell distribution|Maxwell distribution]]), i.e. being always larger than the times of formation of the local hydro
...ction, corresponding to the case of a small deviation from a local Maxwell distribution, is then substituted into these equations. In the zero-th approximation thi
4 KB (525 words) - 20:13, 12 October 2014
...andard one) in terms of the corresponding quantiles of the standard normal distribution, in powers of a small parameter. It was studied by E.A. Cornish and R.A. Fi
is a distribution function depending on $ t $
4 KB (608 words) - 16:41, 15 January 2021
A concept in value-distribution theory. Let $f(z)$ be a meromorphic function in the whole $z$-plane and let $n(r,a
...$f$ to $a$ on $|z|=r$ (cf. [[Value-distribution theory|Value-distribution theory]]). For the majority of values $a$ the quantities $N(r,a,f)$ and $T(r,f)$ a
3 KB (495 words) - 09:04, 26 November 2023
...g to a given probability distribution. In [[Probability theory|probability theory]], attention centres on numerical (that is, scalar) random functions $ X
...-dimensional (vector) random variable characterized by a multi-dimensional distribution function. When $ T $
7 KB (1,062 words) - 08:09, 6 June 2020
Poisson's theorem is a limit theorem in probability theory which is a particular case of the [[Law of large numbers|law of large numbe
...distribution|binomial distribution]] to the [[Poisson distribution|Poisson distribution]]: If $ P _ {n} ( m) $
4 KB (651 words) - 08:06, 6 June 2020
''probability distribution, probability''
(the [[Poisson distribution|Poisson distribution]]);
3 KB (421 words) - 19:48, 8 January 2021
...undaries and knowing the initial step distribution at time $t_0$, the step distribution at $t_0+\Delta t$ is calculated using the mass, momentum and energy balance
...extended to problems of hydrodynamics with heat conduction, to elasticity theory, etc. Owing to an obvious physical interpretation and universality, and sin
3 KB (431 words) - 10:52, 16 April 2014
''in probability theory''
...f the densities of a sequence of distributions to the density of the limit distribution (if the given densities exist), or a classical version of local limit theor
6 KB (856 words) - 15:13, 18 March 2022
...notions of the concept of convergence, of which the most important for the theory of statistical estimation are convergence in probability and convergence wi
be independent random variables with the same normal distribution $ N ( a, \sigma ^ {2} ) $.
4 KB (501 words) - 17:46, 4 June 2020
$#C+1 = 23 : ~/encyclopedia/old_files/data/D110/D.1100030 De Haan theory
...riation, initiated in 1930 by the Yugoslav mathematician J. Karamata. This theory studies asymptotic relations of the form
5 KB (688 words) - 17:32, 5 June 2020
There are theorems for almost-prime numbers that generalize theorems on the distribution of prime numbers in the set of natural numbers. Several [[Additive problems
An example of a result on the distribution of almost-prime numbers generalizing the corresponding one on prime numbers
1 KB (215 words) - 16:10, 1 April 2020
[[Category:Distribution theory]]
...function]] of an [[Infinitely-divisible distribution|infinitely-divisible distribution]]:
8 KB (1,006 words) - 01:14, 19 January 2022
The distribution of the fractional parts $\{\alpha_j\}$ of a sequence of real numbers $\alph
[[Weyl criterion|Weyl's criterion]] (see [[#References|[1]]]) for a distribution modulo one to be uniform: An infinite sequence of fractional parts $\{\alph
3 KB (454 words) - 19:13, 13 November 2023
is the particle distribution function, while the index $ \alpha $
are related to the particle distribution function via
4 KB (530 words) - 08:28, 6 June 2020
...nction. Let $X_1,\ldots,X_n$ be independent random variables with the same distribution such that
2 KB (303 words) - 11:49, 23 November 2023
...ample is drawn from the population at random. From the view of probability theory, a "random" choice means that if the population contains $N$ elements and
...pulation. In so doing it is assumed that the sampling rule is defined by a distribution function $F$, so that the probability of obtaining an experimental result c
3 KB (427 words) - 17:10, 30 December 2018
$#C+1 = 74 : ~/encyclopedia/old_files/data/N067/N.0607460 Normal distribution
[[Category:Distribution theory]]
11 KB (1,515 words) - 08:03, 6 June 2020
...n]]). In (a1), $f$ is the density function of some [[Distribution function|distribution function]] $F$ having all moments $\alpha _ { k } = \int x ^ { k } d F ( x
...yes" , one says that the moment problem has a unique solution, or that the distribution function $F$ is M-determinate. Otherwise, the moment problem has a non-uniq
9 KB (1,390 words) - 07:41, 4 February 2024
A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [[#Refer
valued random element. Its distribution is described by the corresponding [[Probability measure|probability measure
4 KB (571 words) - 16:44, 4 June 2020
...encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350603.png" /> is the distribution density of the random variable
is the density of the standard [[Normal distribution|normal distribution]], and
6 KB (866 words) - 11:09, 12 May 2012
be the number of cells in which, after distribution, there are exactly $ r $
and to study the asymptotic properties of their distribution as $ n , N \rightarrow \infty $.
7 KB (1,068 words) - 16:23, 4 March 2022
A numerical characteristic of the probability distribution of a random variable. In the most general setting, the mathematical expecta
with respect to the probability distribution $ {\mathsf P} _ {X} $
7 KB (945 words) - 11:49, 10 February 2020
[[Category:Distribution theory]]
of the given distribution series. If one restricts to the first few terms of the series (1), one obta
4 KB (578 words) - 11:31, 21 March 2022
...bution]]; [[Density of a probability distribution|Density of a probability distribution]]). Since $ \{ {\eta ( t ) } : {0 \leq t \leq 1 } \} $
frequently appear in [[Probability theory|probability theory]]. In particular, many limit distributions of the Bernoulli excursion $ \
4 KB (575 words) - 06:29, 30 May 2020
...n space of a random vector describing the concentration of its probability distribution around a certain prescribed vector in terms of the second-order moments. Le
called a dispersion ellipsoid of the probability distribution of $ X $
3 KB (375 words) - 16:43, 20 January 2022
''in probability theory''
...expectation of various functions of a random variable $X$ in terms of the distribution of this variable on the set of real numbers are of importance (cf. [[Mathem
7 KB (1,127 words) - 20:19, 27 January 2020
One of the simplest models in [[probability theory]]. A description of an urn model is as follows: Consider some vessel — an
...he Pólya distributions. The negative binomial distribution and the Poisson distribution arise as limit distributions from certain urn models.
2 KB (335 words) - 07:02, 27 May 2023
which automatically holds if the probability distribution of $ X $
is a [[Sufficient statistic|sufficient statistic]] for the distribution of $ X $,
6 KB (857 words) - 07:07, 21 June 2022
is the inverse function of the [[Normal distribution|normal distribution]] with parameters $ ( 0, 1) $.
individually, the asymptotic distribution of $ X $
3 KB (401 words) - 08:27, 6 June 2020
...nd identically distributed random variables $\xi _ { k }$ converges to the distribution of this functional of the Wiener process.
...)$ by finding the explicit distribution of $f ( X _ { n } )$, which is the distribution of $f ( w )$ by the invariance principle.
6 KB (857 words) - 21:45, 15 December 2020
The distribution among the numbers $1,\ldots,m-1$ of those values of $x$ for which the congr
$n>1$, is solvable (or unsolvable) in integers. Questions on the distribution of power residues and non-residues have been studied most fully in the case
4 KB (628 words) - 19:38, 19 December 2014
...unction|transition function]] if and only if there is a stationary initial distribution $ \mu ( A) $
is finite, then a stationary initial distribution always exists, independent of whether the process has discrete $ ( t= 0 ,
4 KB (641 words) - 07:59, 6 June 2020
A chain of equations (hierarchy) for the one-particle, two-particle, etc., distribution functions of a classical statistical system. The functions are defined as f
particle normalized distribution function which satisfies the [[Liouville-equation(2)|Liouville equation]]:
5 KB (743 words) - 10:59, 29 May 2020
...évy inequality]]). In mathematical statistics, to estimate the median of a distribution in terms of independent results of observations $X_1,\dots,X_n$ one uses a
|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
...lopedia/old_files/data/B017/B.0107390 Boundary value problems in potential theory
...sses. It follows, therefore, that the boundary value problems of potential theory are primarily boundary value problems for elliptic equations and systems (c
5 KB (673 words) - 06:29, 30 May 2020
...s of successive subsets [[#References|[a4]]], in which case the asymptotic distribution is determined by recursive application of polynomials. A tree-based method
...pace-efficient recursive procedure for estimating a quantile of an unknown distribution" ''SIAM J. Scient. Statist. Comp.'' , '''4''' (1983) pp. 706–711</TD><
2 KB (281 words) - 17:22, 7 February 2011
...(cf. also [[Density of a probability distribution|Density of a probability distribution]]), and there is much literature on its properties as a statistical estimat
...eedman, P. Diaconis, "On the histogram as a density estimator: $ L_2 $ theory" ''Z. Wahrsch. Verw. Geb.'' , '''57''' (1981) pp. 453–476</TD></TR></t
3 KB (376 words) - 00:30, 11 June 2013
[[Category:Distribution theory]]
of distribution functions (cf. [[Distribution function|Distribution function]]) of one-dimensional random variables such that:
7 KB (1,007 words) - 04:11, 6 June 2020
have a joint normal distribution, then these two coefficients coincide, since in this case
...ribution of the sample information coefficient leads to the problem of the distribution of the sample information. The analysis of the sample information as a meas
3 KB (440 words) - 19:51, 18 January 2024
''imputation (in the theory of games)''
A distribution of the overall gain of all players in a [[cooperative game]] which satisfie
764 bytes (121 words) - 18:18, 9 January 2016
...nce of the variables. To construct the corresponding test one computes the distribution of $ r _ {s} $
one can use tables of the exact distribution (see [[#References|[2]]], [[#References|[4]]]), and when $ n > 10 $
4 KB (638 words) - 09:15, 6 January 2024
In the theory of order statistics the best studied case is the one where the components
are independent random variables having the same distribution, as is assumed hereafter. If $ F ( u) $
12 KB (1,803 words) - 09:08, 10 April 2023
While the modern theory of [[Correlation|correlation]] and [[Regression|regression]] has its roots
...ave a bivariate normal distribution (cf. also [[Normal distribution|Normal distribution]]), $\rho$ is a parameter of the joint density function
4 KB (683 words) - 07:24, 24 March 2023
...rees are equally probable. Then the height of the random tree has the same distribution as $\max(\eta_0,\ldots,\eta_{2n})$ in the Bernoulli excursion. Explicitly:
...he theory of random graphs" J.H. Dshalalow (ed.) , ''Advances in Queueing Theory, Methods, and Open Problems'' , CRC (1995) pp. 45–78</TD></TR>
3 KB (458 words) - 17:54, 26 December 2017
be independent identically-distributed random variables whose distribution function belongs to the family $ H = \{ F ( x) \} $
of all continuous distribution functions on $ \mathbf R ^ {1} $,
7 KB (902 words) - 17:46, 4 June 2020
''in number theory''
...#References|[1]]] to establish criteria for [[Uniform distribution|uniform distribution]] (cf. [[Weyl criterion|Weyl criterion]]).
5 KB (759 words) - 08:29, 6 June 2020
Under various assumptions regarding the distribution of the $ x _ {i} $,
there are exact and approximate expressions for the distribution of the serial correlation coefficients, and of their moments. Serial correl
2 KB (327 words) - 08:24, 20 January 2024
...ite. The Fourier–Stieltjes transform is extensively applied in probability theory, where the non-decreasing function
is continuous on the left; it is called a distribution, and
3 KB (473 words) - 20:16, 16 January 2024
...ability distribution|Probability distribution]]). Suppose that an a priori distribution for $ \theta $
is chosen. One of the fundamental theorems in the asymptotic theory of Bayesian inference (cf. [[Bayesian approach|Bayesian approach]]) is conc
8 KB (1,171 words) - 10:58, 29 May 2020
A concept in probability theory and measure theory. For the most common and most important case of a product of spaces, see th
be a given distribution (or, more generally, a measure) on $ S _ {i} $,
4 KB (573 words) - 17:45, 4 June 2020
...at gives the integral representation of that linear functional. A positive distribution can only exist if $L ( p ) > 0$ for any polynomial $p$ that is positive on
...moments" $m _ { i j } = \langle f _ { i } , f _ { j } \rangle$. Finding a distribution for an integral representation of $L$ on $\mathcal{R}$ is called a strong m
6 KB (888 words) - 18:32, 21 December 2020
...with the property that if the observed value of a [[random variable]], the distribution of which is connected with the tested hypothesis, falls in that region, the
be the tested hypothesis concerning the distribution of a random variable $ X $,
2 KB (331 words) - 19:26, 30 December 2020
...alized Riemann hypothesis in various classical problems in additive number theory. There are also various other improvements of density theorems.
...n="top"> A.F. Lavrik, "A survey of Linnik's large sieve and the density theory of zeros of L-functions" ''Russian Math. Surveys'' , '''35''' : 2 (1980)
3 KB (519 words) - 11:35, 26 March 2023
See also [[Prime number]]; [[Distribution of prime numbers]].
[[Category:Number theory]]
823 bytes (114 words) - 18:31, 5 May 2023
{{MSC|03E04}} ''in set theory''
{{MSC|28A}} ''in measure theory''
5 KB (723 words) - 19:18, 14 November 2023
...90 : ~/encyclopedia/old_files/data/I050/I.0500910 Infinitely\AAhdivisible distribution
[[Category:Distribution theory]]
11 KB (1,489 words) - 09:00, 13 January 2024
''(in probability theory)''
...If the distribution of some pair (say, $\alpha$ and $\beta$) is fixed, the distribution of all other parameters can be uniquely calculated (and is not necessarily
2 KB (374 words) - 10:24, 16 March 2023
A result in the theory of fluctuations in random walks (cf. [[Random walk|Random walk]]). Let $
be independent random variables with the same distribution (cf. [[Random variable|Random variable]]), and let $ S _ {0} = 0 $,
4 KB (549 words) - 18:47, 5 April 2020
...(cf. also [[Density of a probability distribution|Density of a probability distribution]]) is a scalar function $f _ { X } ( X )$ such that:
The matrix variate normal distribution
8 KB (1,193 words) - 17:37, 11 November 2023
...m of constructing similar tests (cf. [[Similar test|Similar test]]) in the theory of statistical hypothesis testing, and the term "Neyman structure" is use
with distribution in the family $ \{ {P _ {e} ^ {T} } : {e \in \Theta _ {0} } \} $.
6 KB (755 words) - 07:28, 24 March 2023
...tion that links a multi-dimensional [[Probability distribution|probability distribution]] function to its one-dimensional margins. Such functions first made their
...e and continuous, and hence a continuous bivariate [[Distribution function|distribution function]] on the unit square, with uniform margins. Furthermore, setting
9 KB (1,407 words) - 17:59, 4 June 2020
The de la Vallée-Poussin theorem on the distribution of prime numbers: Let $\pi(x)$ be the number of primes smaller than $x$; th
...$x$. This theorem demonstrates the correctness of Gauss' hypothesis on the distribution of prime numbers, viz., as $x \rightarrow \infty$,
4 KB (553 words) - 14:03, 17 March 2020
''in analytic number theory''
Five conjectures, formulated by B. Riemann (1876), concerning the distribution of the non-trivial zeros of the [[Zeta-function|zeta-function]]
1 KB (160 words) - 20:17, 18 October 2014
...some kind of statistical regularity (see [[Probability theory|Probability theory]]). One distinguishes random numbers, being generated by a stochastic appar
with distribution function $ F ( z) $;
6 KB (899 words) - 08:09, 6 June 2020
...he kinetic [[Boltzmann equation|Boltzmann equation]] for a single-particle distribution function $ f ( t , r , v ) $,
...ccessive approximations, in which the local [[Maxwell distribution|Maxwell distribution]] $ f _ { \mathop{\rm loc} } ( r , v \mid n , u , \theta ) $,
6 KB (917 words) - 16:43, 4 June 2020
has a given polynomial distribution, characterized by a vector of positive probabilities $ p = ( p _ {1} \dot
a [[Chi-squared distribution| "chi-squared" distribution]] with $ k - 1 $
7 KB (997 words) - 16:43, 4 June 2020
...has to be clearly stated, even in an elementary exposition on probability theory. In the academic literature this point of view was adopted by W. Feller (se
...bility distribution]] of $X$. This measure is uniquely determined by the [[distribution function]] of $X$:
6 KB (904 words) - 20:43, 1 January 2019
..., who pointed out that the development of [[Probability theory|probability theory]] and the extension of its domain of applications had led to the necessity
The basic concepts of probability theory, such as the [[Characteristic function|characteristic function]], the [[Mat
5 KB (804 words) - 08:09, 6 June 2020
in a Euclidean space and let the probability distribution of this vector belong to the parametric family of distributions defined by
...this problem and its constructive description form the base of the general theory of statistical hypothesis testing.
8 KB (1,101 words) - 11:37, 10 February 2020
order probability distributions (cf. [[Probability distribution|Probability distribution]]) $ P = \{ p _ {1} \dots p _ {K} \} $,
between the source probability distribution $ P $
6 KB (919 words) - 22:11, 5 June 2020
...formed data. Such an approach may be easily carried out, and an asymptotic theory associated with other parameters is useful. See [[#References|[a1]]] and [[
...o [[Outlier|Outlier]]). Further, in certain situations, the usual limiting theory based on knowing $ \lambda $
9 KB (1,146 words) - 06:29, 30 May 2020
...n in the sense of the classical theory of functions, and is defined in the theory of generalized functions as a singular [[Generalized function|generalized f
...<0$, $h(x)=1$ for $x>0$ (the value at zero does not matter; as usual for a distribution it suffices for it to be defined apart from a set of measure zero).
2 KB (390 words) - 22:12, 31 December 2018
...precisely, suppose that the random variables $X_1,\dots,X_n$ have a joint distribution in $\R^n$, and let $X^*_{1;3\dots n}$, $X^*_{2;3\dots n}$ be the best linea
[[Student distribution|Student distribution]] with $N-n$ degrees of freedom.
4 KB (537 words) - 20:05, 6 April 2012
''neutron age theory''
...r example, are not permitted). The assumption which underlies neutron flow theory is that the slowing down neutrons lose their energy continuously rather tha
3 KB (514 words) - 11:47, 5 July 2014
which satisfies (1) and is such that the distribution function of $ \mathbf w _ {n} $
converges monotone to the distribution function of $ \mathbf w _ {n} ^ {0} $
18 KB (2,750 words) - 19:29, 16 January 2024
...ounters not the probability integral, but the [[Normal distribution|normal distribution]] function
having the normal distribution with mathematical expectation 0 and variance $ \sigma ^ {2} $,
4 KB (598 words) - 19:23, 11 January 2024
...ntial equation which is satisfied by the [[Potential|potential]] of a mass distribution inside domains occupied by the masses creating this potential. For the [[Ne
where $\rho=\rho(x_1,\dots,x_n)$ is the density of the mass distribution, $\sigma(S^n)=n\pi^{n/2}/\Gamma(n/2+1)$ is the area of the unit sphere $S^n
2 KB (375 words) - 06:54, 27 August 2014
...gence]] in order to study the [[Convergence in distribution|convergence in distribution]] of stochastic processes with jumps.
...ed random variables (cf. [[Random variable|Random variable]]) converges in distribution if and only if their finite-dimensional distributions converge and the laws
4 KB (518 words) - 08:14, 6 June 2020
...sity (cf. [[Density of a probability distribution|Density of a probability distribution]]) of an observable random vector $ X = ( X _ {1} \dots X _ {n} ) $
Property (*) of the uniform distribution for $ R $
4 KB (582 words) - 08:05, 6 June 2020
...or constructing estimators of unknown parameters in statistical estimation theory.
with distribution $ {\mathsf P} _ \theta $
6 KB (728 words) - 17:20, 6 January 2024
be random variables whose joint distribution function $ F ( x , \theta ) $
is used with distribution function $ G ( t , \theta ) $,
8 KB (1,137 words) - 07:29, 24 March 2023
...10/b01511031.png" /> is absolutely continuous, and the density of the mass
distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...cyclopediaofmath.org/legacyimages/b/b015/b015110/b01511046.png" />, a mass
distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
21 KB (2,931 words) - 16:56, 7 February 2011
...This part may be considered as the first serious study ever of probability theory. The book was published in 1713 by N. Bernoulli (a nephew of Jacob Bernoull
...lusively based on a study of the decrease of probabilities in the binomial distribution as one moves away from the most probable value, was accompanied by an inequ
4 KB (524 words) - 10:58, 29 May 2020
has a definite probability distribution whose mathematical expectation is a function of $ x $:
In probability theory, the problem of regression is solved in case the values of the regression v
10 KB (1,524 words) - 08:10, 6 June 2020
...91 : ~/encyclopedia/old_files/data/D031/D.0301110 Density of a probability distribution,
The derivative of the [[Distribution function|distribution function]] corresponding to an absolutely-continuous probability measure.
6 KB (898 words) - 17:32, 5 June 2020
A conjecture on the distribution of [[prime number]]s.
...">[2]</TD> <TD valign="top"> Richard K. Guy, "Unsolved problems in number theory" (3rd ed.) Springer-Verlag (2004) {{ISBN|0-387-20860-7}} {{ZBL|1058.11001}}
1 KB (161 words) - 19:26, 14 November 2023
The [[Probability distribution|probability distribution]] of a [[Brownian motion|Brownian motion]] $ \{ {B ( t ) } : {t \geq 0 }
...e (cf. also [[Constructive quantum field theory|Constructive quantum field theory]]) that can be supported by the space $ C = C [ 0, \infty ) $
5 KB (792 words) - 08:29, 6 June 2020
A proposition from the theory of statistical estimation on which a method for the improvement of unbiased
...er a sufficient statistic. That is how the best unbiased estimator for the distribution function of the normal law is constructed in the following example, which i
7 KB (1,026 words) - 19:47, 16 January 2024
is called weakly convergent to a distribution $ P $
Weak convergence is a basic type of convergence considered in probability theory. It is usually denoted by the sign $ \Rightarrow $.
6 KB (936 words) - 19:36, 5 June 2020
cf. [[Binomial distribution|Binomial distribution]]). Hence $ {\mathsf E} P _ {n} ( C ) = P ( C ) $,
converges in distribution, as $ n \rightarrow \infty $,
9 KB (1,327 words) - 19:37, 5 June 2020
...gree $n$ modulo $p$. See [[Distribution of power residues and non-residues|Distribution of power residues and non-residues]].
...op">[a1]</TD> <TD valign="top"> W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , Springer & PWN (1990) pp. 394ff</TD></TR>
3 KB (435 words) - 19:32, 19 December 2014
...nt which must be estimated from the sampling data). In the latter case the theory is usually employed to study the properties of random objects (for example,
...] and are often employed in sociological studies. According to probability theory, the sample will correctly reproduce the properties of the population as a
11 KB (1,705 words) - 08:12, 6 June 2020
with probability 1, that is, the conditional probability distribution of $ X ( t _ {n+1} ) $
coincides (almost certainly) with the conditional distribution of $ X ( t _ {n+1} ) $
3 KB (412 words) - 19:53, 1 November 2023
A linear connection is determined by a [[Horizontal distribution|horizontal distribution]] on the principal bundle $ P $
The horizontal distribution is determined, moreover, by the differential system $ \omega _ \alpha
6 KB (870 words) - 22:17, 5 June 2020
...1 = 79 : ~/encyclopedia/old_files/data/S087/S.0807320 Statistical decision theory
...n a broader interpretation of the term, statistical decision theory is the theory of choosing an optimal non-deterministic behaviour in incompletely known si
12 KB (1,679 words) - 08:23, 6 June 2020
An approximation of sample quantiles by empirical distribution functions.
...ution function|Distribution function]]; [[Empirical distribution|Empirical distribution]]) of the first $n$ random variables and denote the uniform empirical proce
9 KB (1,217 words) - 16:45, 1 July 2020
''for the unknown true value of a scalar parameter of a probability distribution''
...ies that are functions of the results of observations subject to the given distribution. Let $ X $
6 KB (853 words) - 22:13, 5 June 2020
...and random errors (concerning gross errors see [[Errors, theory of|Errors, theory of]]; in the rest of this article it will be assumed that the observations
The effect of random errors is estimated by methods of the theory of errors. If $ Y _ {1} \dots Y _ {n} $
5 KB (737 words) - 08:07, 6 June 2020
[[Distribution of prime numbers|distribution of prime numbers]] in the natural series.
...bius function associated to a [[partially ordered set]], see [[Enumeration theory]]. In this context, the arithmetic Möbius function defined in this articl
3 KB (427 words) - 08:19, 4 November 2023
with probability distribution $ \{ {p _ {k} } : {1 , 2 ,\dots } \} $,
is the conditional distribution of $ \xi $
7 KB (1,067 words) - 13:06, 6 January 2024
The density hypothesis enables one to obtain results in the theory of prime numbers that are comparable with those following from the Riemann
functions is used in the theory of the distribution of prime numbers belonging to arithmetic progressions.
5 KB (705 words) - 09:02, 6 January 2024
...the distribution of the fractional parts of a wide class of functions, the distribution of prime numbers in series of natural numbers, additive problems such as th
...ences|[5]]]. For the application of Vinogradov's method in analytic number theory see [[#References|[1]]], [[#References|[2]]], , [[#References|[5]]], [[#Ref
9 KB (1,606 words) - 09:42, 9 November 2014
be random variables with given joint distribution, let $ m _ {X} $
A natural index of the concentration of the distribution near the regression curve $ y ( x) $
12 KB (1,686 words) - 04:11, 15 June 2022
...have a probability distribution close to the [[Normal distribution|normal distribution]].
The distribution functions
20 KB (2,992 words) - 07:15, 21 March 2023
...hat the transfer for the fibres of a vector bundle defined by a horizontal distribution ceases to have a linear character, that is, is not a linear isomorphism of
...le, a [[Kawaguchi space|Kawaguchi space]]). The foundations of the general theory of non-linear connections are fairly well developed and applications of som
7 KB (1,089 words) - 12:15, 18 February 2022
...ory, proposed by L. Boltzmann for the determination of the single-particle distribution function of an ideal mono-atomic gas [[#References|[1]]]. In dimensionless
Here $f(x,v,t)$ is the density of the distribution function of the number of particles in the phase space $x\otimes v$, $x$ is
3 KB (545 words) - 15:08, 14 February 2020
...h a process is called a renewal process (cf. also [[Renewal theory|Renewal theory]]) and objects of interest are the behaviour for large (time) $t$:
...e mathematical theory of such processes is called [[Renewal theory|renewal theory]] and Blackwell's renewal theorem plays a central role in it. See [[#Refere
10 KB (1,526 words) - 19:03, 23 January 2024
...ble|Random variable]]) are independent identically distributed with common distribution function $ {\mathsf P} ( A ) $
is the empirical distribution, can be represented by a linear combination of $ U $-
9 KB (1,249 words) - 08:27, 6 June 2020
is positive, the distribution of the sequence $ \{ {q _ {n+} k } : {k \geq 0 } \} $
of queue lengths for a loss system will converge to the distribution of the stationary sequence $ \{ {q ^ {k} } : {k \geq 0 } \} $
10 KB (1,560 words) - 08:02, 14 January 2024
* Bugeaud, Yann. "Distribution modulo one and Diophantine approximation", Cambridge Tracts in Mathematics
* Berthé, Valérie; Rigo, Michel. "Combinatorics, automata, and number theory". Encyclopedia of Mathematics and its Applications '''135''' Cambridge Univ
997 bytes (135 words) - 19:33, 6 December 2023
...M. Legendre]] used the ideas of this sieve to derive an estimate for the [[distribution of prime numbers]] of the form $\pi(x) < x/\log\log x$.
...yond", ''Sieve Methods, Exponential Sums, and Their Applications in Number Theory'',
1 KB (220 words) - 10:32, 30 March 2024
are the reduced distribution functions
are the distribution functions in the phase space at time $ t $,
6 KB (842 words) - 17:31, 5 June 2020
is a random variable with a [[Normal distribution|normal distribution]]. Zones of normal attraction have been studied for the case
also form part of large deviation theory if the main part of these expectations for large values of $ a $
3 KB (451 words) - 20:13, 10 January 2024
...cted component of the interior of $K$, i.e. the problem on the equilibrium distribution of an electric charge $\lambda(K)$ on the surface $S$ of a conductor $K$.
...'s problem, have a constant value on $K$ (see [[Potential theory|Potential theory]], and also [[#References|[2]]]). The solution $\nu(x)$ for the problem \eq
5 KB (913 words) - 15:34, 14 February 2020
...ve problems of number theory, cf. [[Additive number theory|Additive number theory]]).
In additive problems of number theory, trigonometric sums occur in the following way.
10 KB (1,485 words) - 14:56, 7 June 2020
...stituents of systems for the transmission of information considered in the theory of information transmission. It is used in the mathematical description of
gives the conditional probability distribution of the signal received by the receiver under the condition that the transmi
10 KB (1,309 words) - 17:45, 4 June 2020
...thods of the theory of statistical estimation form the basis of the modern theory of errors; physical constants to be measured are commonly used as the unkno
...al mean (1) is the best statistical estimator. However, if the probability distribution of the random variables $ X _ {i} $
39 KB (5,647 words) - 16:41, 13 January 2024
cf. [[Density of a probability distribution|Density of a probability distribution]]) with densities $ p ( \omega ; t ) $
...accuracy of decision rules in the problem of estimating the parameter of a distribution law. The variance of any unbiased estimator $ \tau ( \omega ) = \tau ( \o
6 KB (792 words) - 07:58, 14 January 2024
The study of adaptive algorithms is generally made through the theory of [[Stochastic approximation|stochastic approximation]].
where $H$ is known, but the distribution of $X$ is unknown and may or may not depend on $\theta$ (as indicated by th
8 KB (1,242 words) - 15:31, 1 July 2020
...hod of trigonometric sums and by methods of algebraic and geometric number theory.
...ic number theory, Diophantine approximation, computational geometry, graph theory, integral geometry, and other areas, see [[#References|[a1]]].
2 KB (239 words) - 19:03, 2 October 2016
[[Number theory|number theory]] in which ideas and methods from
[[Probability theory|probability theory]] are used.
10 KB (1,718 words) - 04:55, 8 August 2018
...to probabilities and therefore the [[Probability distribution|probability distribution]] for a molecule over its possible states can be introduced. Let the set of
This is called the Boltzmann distribution.
9 KB (1,357 words) - 16:59, 1 July 2020
...s|[a1]]] or [[#References|[a6]]] for a general reference and also [[Number theory, probabilistic methods in]]). One of the main theorems in this area, and a
...$\log\log n$ and standard deviation $\sqrt{\log\log n}$ (cf. also [[Normal distribution]]).
4 KB (647 words) - 07:30, 18 March 2023
then the distribution of $ \beta _ {n} $
is the exponent of the distribution of $ \tau _ {j} ^ {s} $.
24 KB (3,620 words) - 16:42, 30 December 2020
...op">[1]</TD> <TD valign="top"> M.G. Kendall, A. Stuart, "The advanced theory of statistics" , '''3. Design and analysis, and time series''' , Griffin (
...ndependently and identically distributed with [[Normal distribution|normal distribution]], that is, the so-called standard linear multiple regression model or, bri
3 KB (484 words) - 18:22, 18 December 2020
...nction $\pi(x)$ — the number of primes not larger than $x$ — describes the distribution of primes; $\pi(x,q,l)$ gives the number of primes not larger than $x$ in t
Algebraic number theory deals with generalizations of the above arithmetic functions of a natural a
4 KB (608 words) - 08:18, 4 November 2023
* the fact that the distribution is completely summarized by a relatively small number of parameters (which
.... There may, however, be problems associated with the extreme tails of the distribution, and these tails are particularly important
9 KB (1,305 words) - 09:48, 28 October 2023
''in probability theory''
...tablished by J. Bernoulli (1713) and P. Laplace (1812), are related to the distribution of the deviation of the frequency $ \mu _ {n} /n $
14 KB (2,087 words) - 22:16, 5 June 2020
...cf. [[Measure|Measure]]) on $k$ (cf. also [[Binomial distribution|Binomial distribution]]). Here, with $p = e ^ { \theta } / ( 1 + e ^ { \theta } )$ and $q = 1 - p
...{ 2 }$ are both unknown parameters (cf. also [[Normal distribution|Normal distribution]]). Here, $\nu ( d \omega ) = d x / \sqrt { 2 \pi }$, the space $E$ is $\ma
12 KB (1,855 words) - 17:45, 1 July 2020
...istics of objects. As a rule, these are integral equations for the unknown distribution functions of the true characteristics. For example, an important equation f
is the distribution function $ \Delta ( r ) $
9 KB (1,236 words) - 08:23, 6 June 2020
dimensional Euclidean space. The distribution of $ X _ {t} $
...and asymptotically normally distributed (cf. [[Normal distribution|Normal distribution]]) with minimum size asymptotic confidence zones (cf. [[Confidence estimati
5 KB (684 words) - 18:48, 5 April 2020
...n="top">[2]</TD> <TD valign="top"> H. Davenport, "Multiplicative number theory" , Springer (1980)</TD></TR>
...op">[3]</TD> <TD valign="top"> D.R. Heath-Brown, S.I. Patterson, "The distribution of Kummer sums at prime arguments" ''J. Reine Angew. Math.'' , '''310'''
1 KB (211 words) - 20:25, 31 October 2014
...act, Hilbert advanced the hypothesis according to which the problem of the distribution of prime numbers allows one to solve both the Goldbach–Euler problem and
[[Category:Number theory]]
1 KB (201 words) - 20:18, 14 October 2014
is specified, then, according to the general theory of statistical hypotheses testing, it is precisely $ H _ {1} $
Tables of the "chi-squared" distribution yield the critical value $ \chi _ {1} ^ {2} ( 0.05 ) = 3.841 $
5 KB (621 words) - 08:13, 6 June 2020
...ipal problems of astrophysics is the structure and evolution of stars. The theory of the internal structure of stars leads to differential equations which de
...in the nebulae and in the interstellar space is based on the mathematical theory of radiative transfer, which was intensively developed in astrophysics. In
8 KB (1,204 words) - 18:48, 5 April 2020
...><TD valign="top">[2]</TD> <TD valign="top"> V.V. Nemytskii, "On the orbit theory of general dynamic systems" ''Mat. Sb.'' , '''23 (65)''' : 2 (1948) pp. 161
define a distribution on $ U $.
5 KB (702 words) - 17:45, 4 June 2020
$#C+1 = 34 : ~/encyclopedia/old_files/data/R081/R.0801250 Renewal theory
...renewal of the elements of some system. The principal concepts in renewal theory are those of a renewal process and renewal equation. A renewal process may
4 KB (599 words) - 08:10, 6 June 2020
''in kinetic gas theory''
...l equation which approximately describes the evolution of the one-particle distribution function of a sufficiently-rarefied gas without internal degrees of freedom
4 KB (508 words) - 10:59, 29 May 2020
$#C+1 = 59 : ~/encyclopedia/old_files/data/E036/E.0306240 Errors, theory of
...random errors. The basic problems of the theory of errors are to study the distribution laws of random errors, to seek estimates (see [[Statistical estimator|Stati
12 KB (1,805 words) - 08:11, 21 January 2024
...Vinogradov–Bombieri theorem on the average [[Distribution of prime numbers|distribution of prime numbers]] in arithmetic progressions also gives a solution of the
[[Category:Number theory]]
3 KB (485 words) - 19:43, 5 June 2020
has exponential distribution with unknown shift parameter $ \theta $,
has normal distribution $ N( \theta , 1) $
4 KB (549 words) - 19:05, 17 January 2024
...dots\}$ is the strategy (cf. [[Strategy (in game theory)|Strategy (in game theory)]]) of the player. If the player terminates the game at the moment $t$, his
The theory of games of chance is part of the general theory of controlled stochastic processes (cf. [[Controlled stochastic process|Con
2 KB (382 words) - 00:29, 25 November 2018
According to the central limit theorem (''CLT''), the distribution function $F_n$ of a normalized sum
...egorical data. Applying the ''CLT'' to a binomial random variable $T$ with distribution $B(n,p)$, with mean $np$ and variance $npq (q=1-p)$, the normal approximati
11 KB (1,719 words) - 20:18, 12 March 2016
In contrast to classical information, described by means of a probability distribution on the space of signals $ X $,
...als. Transmission of information is described, as in classical information theory, by the scheme
6 KB (890 words) - 08:08, 6 June 2020
...f sums of independent random variables to the [[Normal distribution|normal distribution]]. The precise statement of Lyapunov's theorem is as follows: Suppose that
...lity of the [[Central limit theorem|central limit theorem]] of probability theory. Later, conditions were established that extend Lyapunov's conditions and t
9 KB (1,266 words) - 14:20, 14 August 2023
In the literature on [[Queueing theory|queueing theory]] other special forms of service systems are discussed. However, one must k
...mental" systems; underlying these methods is the apparatus of probability theory, both in its general aspects and in specially developed aspects.
15 KB (2,349 words) - 09:05, 21 January 2024
...is called mean-power approximation (with exponent $q$) with respect to the distribution $d\sigma(t)$. If $\sigma(t)$ is absolutely continuous and $\phi(t)=\sigma(t
...align="top"> J.R. Rice, "The approximation of functions" , '''1. Linear theory''' , Addison-Wesley (1964)</TD></TR></table>
2 KB (331 words) - 08:01, 23 August 2014
...= 222 : ~/encyclopedia/old_files/data/V096/V.0906020 Value\AAhdistribution theory,
''Nevanlinna theory''
20 KB (2,984 words) - 08:27, 6 June 2020
...(single-layer or double-layer), a volume potential, etc. (see [[Potential theory]]).
1 KB (222 words) - 17:06, 27 December 2017
Kronecker's theorem can be derived from the [[Duality|duality]] theory for commutative topological groups (cf. [[Topological group|Topological gro
is even uniformly distributed, cf. [[Uniform distribution|Uniform distribution]].
4 KB (615 words) - 22:15, 5 June 2020
theory to data.
rules, often were introduced in astronomy. The theory of weighted least
8 KB (1,114 words) - 06:48, 8 October 2023
$#C+1 = 148 : ~/encyclopedia/old_files/data/M063/M.0603710 Metric theory of numbers
...tic properties. Metric number theory is closely connected with probability theory, which sometimes proves an opportunity to use its methods and results in th
19 KB (2,729 words) - 08:05, 14 January 2024
...) play an important part in [[Value-distribution theory|value-distribution theory]].
...ation of the formulas (1)–(3) has been obtained by M.M. Dzhrbashyan in his theory of classes of meromorphic functions (see {{Cite|Dz}}). He succeeded in obta
7 KB (1,197 words) - 20:51, 19 April 2012
has a given probability distribution at a fixed value $ x $
...atter case, the analysis is substantially supplemented by methods from the theory of [[Correlation (in statistics)|correlation (in statistics)]]).
12 KB (1,774 words) - 13:04, 13 January 2024
...nary codes whose distance distribution is the MacWilliams transform of the distribution of the Delsarte–Goethals codes, see [[#References|[a4]]]. These codes act
...op">[a5]</td> <td valign="top"> F.J. MacWilliams, N.J.A. Sloane, "The theory of error-correcting codes" , North-Holland (1977)</td></tr></table>
4 KB (592 words) - 16:52, 1 July 2020
...certain initial capital $u$ at its disposal. One important problem in risk theory is to investigate the ruin probability $\Psi(u)$, i.e., the probability tha
...dom variable]]) $X_1,X_2,\dots$, having the common [[Distribution function|distribution function]] $F$, with $F(0)=0$, and mean value $\mu$; it is assumed that $h(
4 KB (698 words) - 13:17, 3 October 2014
with probability distribution $ p _ {m} = {\mathsf P} \{ \xi = m \} $,
is an arbitrary probability distribution on $ Y $,
6 KB (837 words) - 11:58, 7 January 2024
...es necessary and sufficient conditions for the asymptotic normality of the distribution function of sums of independent random variables that have finite variances
...><TR><TD valign="top">[3]</TD> <TD valign="top"> M. Loève, "Probability theory" , Springer (1977)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"
2 KB (317 words) - 17:05, 13 January 2024
...rywhere, for all $\theta\in\Theta$. For example, a family of [[exponential distribution]]s is complete. If the relation \eqref{eq:a1} is satisfied under the furthe
<TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Zacks, "The theory of statistical inference" , Wiley (1971)</TD></TR>
2 KB (237 words) - 12:23, 12 November 2023