Редактирование: Статистические характеристики дискретных сред

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== Терминология ==

* '''Начальным''' и '''центральным''' моментом [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 случайной величины] <math>\displaystyle X</math> называются, соответственно, величины
:: <math>\nu_k = \mathbb{E}\left[X^k\right], \qquad \mu_k = \mathbb{E}\left[(X - \mathbb{E}X)^k\right]</math>
: где <math>\mathbb{E}</math> — [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 математическое ожидание] случайной величины, <math>k</math> — степень момента.
* <math>\nu_1=\mathbb{E}\left[X\right]</math> — [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 математическое ожидание], <math>\mu_2=\sigma^2</math> — [https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B дисперсия], <math>\mu_3/\sigma^3</math> — [https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%B0%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B8 коэффициент асимметрии], <math>(\mu_4/\sigma^4-3)</math> — [https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D1%81%D1%81 коэффициент эксцесса].

== Словарь ==

* [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 Случайная величина] — [https://en.wikipedia.org/wiki/Random_variable Random variable]
* [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 Математическое ожидание] — [https://en.wikipedia.org/wiki/Expected_value Expected value] (mathematical expectation)
* [https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B Дисперсия случайной величины] — [https://en.wikipedia.org/wiki/Variance Variance]
* [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%82%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BA%D0%BB%D0%BE%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5 Среднеквадратическое отклонение] — [https://en.wikipedia.org/wiki/Standard_deviation Standard deviation]
* [https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%B0%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B8 Коэффициент асимметрии] — [https://en.wikipedia.org/wiki/Skewness Skewness]
* [https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%86%D0%B5%D1%81%D1%81 Коэффициент эксцесса] — [https://en.wikipedia.org/wiki/Kurtosis Kurtosis] 

== Таблица ==

{| class="wikitable"
|-
! Обозначение
! Русское название
! English name
|-
| <math>X</math>
| [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 Случайная величина]
| [https://en.wikipedia.org/wiki/Random_variable Random variable]
|-
| <math>F(x) = \mathbb{P}( X \leqslant x )</math>
| [https://ru.wikipedia.org/wiki/%D0%9A%D1%83%D0%BC%D1%83%D0%BB%D1%8F%D1%82%D0%B8%D0%B2%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F Функция распределения]
| [https://en.wikipedia.org/wiki/Cumulative_distribution_function Cumulative distribution function]
|-
| <math>f(x) = F'(x)</math>
| [https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8 Плотность распределения]
| [https://en.wikipedia.org/wiki/Probability_density_function Probability density function] (distribution density)
|-
| <math>\left<X\right> = \int_{-\infty}^\infty x f(x)dx</math>
| [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D0%B6%D0%B8%D0%B4%D0%B0%D0%BD%D0%B8%D0%B5 Математическое ожидание]
| [https://en.wikipedia.org/wiki/Expected_value Expected value] (mathematical expectation)
|-
| <math>\varphi(s) = \left<e^{isX}\right> = \int_{-\infty}^\infty e^{isx} f(x)dx</math>
| [https://ru.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%B0%D0%BA%D1%82%D0%B5%D1%80%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B Характеристическая функция]
| [https://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29 Characteristic function]
|-
| <math>M(s) = \left<e^{sX}\right> = \int_{-\infty}^\infty e^{sx} f(x)dx</math>
| [https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D1%8F%D1%89%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D0%BE%D0%B2 Производящая функция моментов]
| [https://en.wikipedia.org/wiki/Moment-generating_function Moment-generating function]
|-
| <math>\nu_n = \left<X^n\right> = \int_{-\infty}^\infty x^n f(x)dx = \left.\frac{d^n}{ds^n} M(s)\right|_{s=0}</math>
| Начальный момент [https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%8B_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B]
| Raw moment [https://en.wikipedia.org/wiki/Moment_%28mathematics%29]
|-
| <math>\mu_n = \left<(X-\nu_1)^n\right> = \int_{-\infty}^\infty (x-\nu_1)^n f(x)dx</math>
| Центральный момент [https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%8B_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B]
| Central moment [https://en.wikipedia.org/wiki/Moment_%28mathematics%29]
|-
| <math>\kappa_n = \left.\frac{d^n}{ds^n}\ln M(s)\right|_{s=0} = (-i)^n\left.\frac{d^n}{ds^n}\ln \varphi(s)\right|_{s=0}</math>
| [https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%83%D0%B8%D0%BD%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82 Полуинвариант] (кумулянт)
| [https://en.wikipedia.org/wiki/Cumulant Cumulant]
|}

== Ссылки ==

* Случайные величины и их характеристики
** [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0 Случайная величина]
** [https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE Вероятностное пространство]
** [https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%8B_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%BE%D0%B9_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D1%8B Моменты случайной величины]
** [https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81 Случайный процесс]
** [https://en.wikipedia.org/wiki/Random_vector Multivariate random variable] (random vector)

* Науки
** [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 Статистическая механика]
*** [https://ru.wikipedia.org/w/index.php?title=%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%9B%D0%B8%D1%83%D0%B2%D0%B8%D0%BB%D0%BB%D1%8F&redirect=no Уравнение Лиувилля]
*** [https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BF%D0%BE%D1%87%D0%BA%D0%B0_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B9_%D0%91%D0%BE%D0%B3%D0%BE%D0%BB%D1%8E%D0%B1%D0%BE%D0%B2%D0%B0 Цепочка уравнений Боголюбова]
** [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0 Статистическая физика]
** [https://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B7%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BA%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%D0%B0 Физическая кинетика]
*** [https://ru.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0 Кинетическое уравнение Больцмана]
** [https://ru.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BA%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%D0%B0 Химическая кинетика]
*** [https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%90%D1%80%D1%80%D0%B5%D0%BD%D0%B8%D1%83%D1%81%D0%B0 Уравнение Аррениуса]

* Discrete calculus and discrete analysis
** [https://en.wikipedia.org/wiki/Finite_difference Finite difference]
** [https://en.wikipedia.org/wiki/Recurrence_relation#Relationship_to_difference_equations_narrowly_defined Recurrence relation]
** [https://en.wikipedia.org/wiki/Logistic_map Logistic map]
** [https://en.wikipedia.org/wiki/Z-transform Z-transform]
** See also [https://en.wikipedia.org/wiki/Discrete_mathematics#Calculus_of_finite_differences.2C_discrete_calculus_or_discrete_analysis Discrete mathematics]

== Литература ==

*Борн М. «Непрерывность, детерминизм, реальность» в книге «Размышления и воспоминания физика». М.: Мир, 1977. стр.162-187. (Скачать djvu: [http://padabum.com/x.php?id=28958 2.38 Mb], [http://padabum.com/d.php?id=28958 страница для скачивания]).
:Born M. «Continuity, determinism and reality», Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser, Bind 30, Nr.2, (1955) 1-26.  
:— Впервые рассмотрена (согласно [https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0]) классическая статистическая механика одной частицы (1955 г.) 

* Лукач Е. Характеристические функции. Пер. с анг. 1979. М.: Наука. 424 с. [http://urss.ru/cgi-bin/db.pl?lang=Ru&blang=ru&page=Book&id=24313#FF1 Оглавление]  
:Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Downlod djvu: 3.9 Mb, [http://lib.org.by/info/M_Mathematics/MV_Probability/Lukacs%20E.%20Characteristic%20functions%20%282ed.,%20Griffin,%201970%29%28KA%29%28600dpi%29%28T%29%28360s%29_MV_.djvu download page]).
:— A negative result (Theorem 7.3.5): '''The [https://en.wikipedia.org/wiki/Cumulant#cite_ref-3 cumulant] generating function cannot be a finite-order polynomial of degree greater than 2.''' <toggledisplay status=hide showtext="Clarification &gt;&gt;" hidetext="Clarification &lt;&lt;" linkstyle="font-size:default"> (Given the results for the cumulants of the [https://en.wikipedia.org/wiki/Normal_distribution normal distribution], it might be hoped to find families of distributions for which κ<sub>''m''</sub>&nbsp;=&nbsp;κ<sub>''m''+1</sub>&nbsp;=&nbsp;...&nbsp;=&nbsp;0 for some ''m''&nbsp;>&nbsp;3, with the lower-order cumulants (orders 3 to ''m''&nbsp;&minus;&nbsp;1) being non-zero. From the theorem it follows that there are no such distributions.)</toggledisplay>  <toggledisplay status=hide showtext="Origin &gt;&gt;" hidetext="Origin &lt;&lt;" linkstyle="font-size:default"> Данное утверждение является следствием теоремы, впервые доказанной [https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%80%D1%86%D0%B8%D0%BD%D0%BA%D0%B5%D0%B2%D0%B8%D1%87,_%D0%AE%D0%B7%D0%B5%D1%84 Юзефом Марцинкевичем], польским математиком, погибшим во время Второй мировой войны: Marcinkiewicz, J. (1938). Sur une propriete de la loi de Gauss. Math. Zeitschr., 44, 612-618 ([http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN266833020_0044&DMDID=DMDLOG_0058&LOGID=LOG_0058&PHYSID=PHYS_0616 read online], download pdf: 397 Kb [http://gdz.sub.uni-goettingen.de/index.php?id=14&PPN=PPN266833020_0044&DMDID=DMDLOG_0059&LOGID=LOG_0059&PHYSID=PHYS_0618&L=0 download page]). Reprinted in J. Marcinkiewicz, Collected Papers. Panstwowe wydawnictwo Naukowe Warszawa, 1964. [http://zbmath.org/?q=an:0124.24103 Abstract.] </toggledisplay>

* [http://eqworld.ipmnet.ru/ru/library/mathematics/probability.htm Теория вероятностей и математическая статистика] на сайте [http://eqworld.ipmnet.ru/indexr.htm EqWorld]
* [http://lib.org.by/_djvu/M_Mathematics/MV_Probability/ Probability] на сайте [http://lib.org.by/ Белорусская научная библиотека]


{{#ifgroup:sysop|

== Приложение ==

Рассмотрим одномерную дискретную среду, сотоящую из <math>N</math> частиц. Обозначим <math>u_n</math> — некоторую характеристику частицы, например ее перемещение. Введем среднее значение характеристики как 

: <math>\left<u_n\right> = \sum_{n=1}^Nu_n</math>

и среднее значение степени <math>k</math>

: <math>\left<u_n^k\right> = \sum_{n=1}^Nu_n^k</math>.

Если интерпретировать <math>u_n</math> как случайную величину, то при достаточно большом <math>N</math> величину <math>\left<u_n^k\right></math> можно называть <math>k</math>-м моментом случайной величины.

}}