累积分布函数

此條目需要擴充。 (2013年10月26日)
请協助改善这篇條目，更進一步的信息可能會在討論頁或扩充请求中找到。请在擴充條目後將此模板移除。

累积分布函数（英語：cumulative distribution function，CDF）或概率分布函数，简称分布函数，是概率密度函數的积分，能完整描述一個實随机变量 $X$ 的概率分佈。

在標量連續分佈的情況下，它給出了從負無窮到 $x$ 的概率密度函數下的面積。累積分佈函數也用於指定多元隨機變量（英语：Multivariate random variable）的分佈。

定義

對於所有實數值的随机变量 $X$ ，累积分布函数定義如下^[1]^{:p. 77}：

F_{X}(x)=\operatorname {P} (X\leq x)

Eq.1

其中右侧表示随机变量 $X$ 取值小于或等于 $x$ 的概率。

對於 $X$ 位于半闭区间 $(a,b]$ 的概率，其中 $a<b$ ，因此定義是^[1]^{:p. 84}:

\operatorname {P} (a<X\leq b)=F_{X}(b)-F_{X}(a)

Eq.2

在上面的定義中，“小於或等於”符號“≤”是一種約定，不是普遍使用的（例如匈牙利文獻使用“<”），但這種區別對於離散分佈很重要。二項式分布和泊松分布的表格的正確使用取決於此約定。此外，像數學家保羅·皮埃爾·萊維(Paul Lévy)的特徵函數反演公式等重要公式也依賴於“小於或等於”公式。

性質

有界性^[2]
- $\lim _{x\to -\infty }F_{X}(x)=0$
- $\lim _{x\to +\infty }F_{X}(x)=1$
單調性：
- $F_{X}(x_{1})\leq F_{X}(x_{2}),\ {\mbox{if}}\,x_{1}<x_{2}$
右連續性：
- $\lim _{x\rightarrow x_{0}^{+}}F_{X}(x)=F_{X}(x_{0})$

$X$ 之值落在一區間 $(a,b]$ 之內的機率為

\operatorname {P} (a<X\leq b)=F_{X}(b)-F_{X}(a)

一隨機變數 $X$ 的CDF與其PDF的關係為

F_{X}(x)=\int _{-\infty }^{x}f_{X}(t)\,dt

反函数

若累积分布函数 $F$ 是连续的严格增函数，则存在其反函数 $F^{-1}(y),y\in [0,1]$ 。累积分布函数的反函数可以用来生成服从该随机分布的随机变量。设若 $F_{X}(x)$ 是概率分布 $X$ 的累积分布函数，并存在反函数 $F_{X}^{-1}$ 。若 $a$ 是 $[0,1)$ 区间上均匀分布的随机变量，则 $F_{X}^{-1}(a)$ 服从 $X$ 分布。

互补累积分布函数

互补累積分布函数（complementary cumulative distribution function、CCDF），是对连续函数，所有大于 $a$ 的值，其出现概率的和。

F(a)=P(x>a)

參見

參考

^ ^1.0 ^1.1 Park, Kun Il. Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. 2018. ISBN 978-3-319-68074-3.
^ 《概率論與數理統計教程》茆詩松程依明濮曉龍

查论编概率分布的理论
概率质量函数(pmf) 概率密度函数(pdf) 累积分布函数(cdf) 分位函数
矩中心矩期望值方差標準差偏度峰度
矩生成函數(mgf) 特征函数概率生成函数(pgf) 累积量

这是一篇关于数学的小作品。您可以通过编辑或修订扩充其内容。

Read other articles:

Galeberan

Galeberan Setipinna taty Setipinna tatyStatus konservasiRisiko rendahIUCN98990439 TaksonomiKerajaanAnimaliaFilumChordataKelasActinopteriOrdoClupeiformesFamiliEngraulidaeGenusSetipinnaSpesiesSetipinna taty Valenciennes, 1848 lbs Galeberan (Setipinna taty) adalah spesies ikan bersirip pari dalam famili Engraulidae. Ini ditemukan di perairan pantai dan muara di wilayah Indo-Pasifik barat tropis. Referensi Artikel bertopik ikan ini adalah sebuah rintisan. Anda dapat membantu Wikipedia dengan meng...

Counterweight

Equivalent weight that balances a mechanical system For other uses, see Counterweight (disambiguation). A simple crane A counterweight is a weight that, by applying an opposite force, provides balance and stability of a mechanical system.[1] The purpose of a counterweight is to make lifting the load faster and more efficient, which saves energy and causes less wear and tear on the lifting machine.[2] Counterweights are often used in traction lifts (elevators), cranes and funfa...

Asakusa Shrine

Shinto shrine in Asakusa district, Tokyo, Japan Asakusa Shrine浅草神社ReligionAffiliationShintoDeityHinokuma HamanariHinokuma TakenariHajino Matsuchi[1]Ebisu[2]LocationLocation2-26-1, Asakusa, Taitō-kuTokyo 111-0032Shown within JapanGeographic coordinates35°42′54.50″N 139°47′50.77″E / 35.7151389°N 139.7974361°E / 35.7151389; 139.7974361ArchitectureStyleGongen-zukuriFounderTokugawa IemitsuDate established1649Websitewww.asakusajinja.jp/en...

Ukraine and weapons of mass destruction

Weapons of mass destruction By type Biological Chemical Nuclear Radiological By country Albania Algeria Argentina Australia Brazil Bulgaria Canada China Egypt France Germany India Iran Iraq Israel Italy Japan Libya Mexico Myanmar Netherlands North Korea Pakistan Philippines Poland Rhodesia Romania Russia (Soviet Union) Saudi Arabia South Africa South Korea Spain Sweden Switzerland Syria Taiwan Ukraine United Kingdom United States Proliferation Chemical Nuclear Missiles Treaties List of treat...

2016年美國總統選舉

2016年美國總統選舉 ← 2012 2016年11月8日 2020 → 538個選舉人團席位獲勝需270票民意調查投票率55.7%[1][2] ▲ 0.8 % 获提名人唐納·川普希拉莉·克林頓政党共和黨民主党家鄉州紐約州紐約州竞选搭档迈克·彭斯蒂姆·凱恩选举人票 304[3][4][註 1] 227[5] 胜出州/省 30 + 緬-2 20 + DC 民選得票 62,984,828[6] 65,853,514[6]...

The Battle at Elderbush Gulch

The Battle at Elderbush GulchPoster filmSutradaraD. W. GriffithDitulis olehD. W. GriffithHenry Albert PhillipsPemeranMae MarshLillian GishSinematograferG. W. BitzerDistributorGeneral Film CompanyTanggal rilis November 1913 (1913-11) Durasi29 menitNegaraAmerika SerikatBahasaBisu dengan intertitel Inggris Film lengkap The Battle at Elderbush Gulch (juga dikenal sebagai The Battle of Elderbush Gulch) adalah sebuah film koboi bisu Amerika Serikat tahun 1913 garapan D. W. Griffith[1] ...

List of Latin place names in Britain

Latin Place Names By country Africa Asia Europe Balkans Britain Continental and Ireland Iberia Italy and Malta Serbia By type Cities Countries Exonyms Mountains Regions Rivers vte This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: List of Latin place names in Britain – news · newspapers · books · scholar · JST...

Antrim, New Hampshire

Town in New Hampshire, United StatesAntrim, New HampshireTownAntrim Town Hall SealLocation in Hillsborough County, New HampshireCoordinates: 43°01′51″N 71°56′20″W / 43.03083°N 71.93889°W / 43.03083; -71.93889CountryUnited StatesStateNew HampshireCountyHillsboroughIncorporatedMarch 22, 1777VillagesAntrimAntrim CenterClinton VillageNorth BranchGovernment • SelectboardMichael Ott, ChairDonna HansonBob Edwards • Town Admi...

Jin dynasty (266–420)

Chinese dynasty For other uses, see Jin. Jin晉8 February 266–10 July 420300XianbeiTashtykKokelKhotanGaojuTurksCHAM-PASargatHYMYARJINDYNASTYGOGU-RYEOWESTERNSATRAPSVAKA-TAKASKUSHANO-SASANIANSLITTLEKUSHANSXIONITESKANGJUSASANIANEMPIREROMANEMPIREHUNSFUNANJushiTOCHARIANSTUYUHUNPaleo-SiberiansSamoyedsTungusMEROËAKSUMThe Jin dynasty and contemporary polities c. 300The Jin dynasty (yellow) at its greatest extent, c. 280, during the Western Jin dynastyCapitalLuoyang (266–311)Chang'an ...

Oil well

Well drilled to extract crude oil and/or gas This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Oil well – news · newspapers · books · scholar · JSTOR (December 2023) (Learn how and when to remove this message) The pumpjack, such as this one located south of Midland, is a common sight in West Texas An oil well ...

Algebraic variety

Mathematical object studied in the field of algebraic geometry This article is about algebraic varieties. For the term variety of algebras, see Variety (universal algebra). The twisted cubic is a projective algebraic variety. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generaliz...

Mashed potato

Potato dish This article is about the food. For other uses, see Mashed potato (disambiguation). See also: Instant mashed potatoes This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Mashed potato – news · newspapers · books · scholar · JSTOR (October 2014) (Learn how and when to remove this message) Mashed potat...

Southern University at Shreveport

Historically black junior college in Louisiana, US Not to be confused with Southern University College or Southern University Southern University at ShreveportSeal of Southern University at ShreveportMottoOpportunity Starts HereTypePublic, HBCU, Community collegeEstablished1967PresidentDr. Ray BeltonStudents3,014LocationShreveport, Louisiana, United StatesColorsBlue and Gold NicknameJaguars and Lady JaguarsSporting affiliationsNJCAA – Miss-Lou ConferenceWebsitewww.susla.edu Sout...

Maresha

Archaeological site in southern Israel Caves of MareshaUNESCO World Heritage SiteLocationShfela, Israel,Part ofCaves of Maresha and Bet-Guvrin in the Judean Lowlands as a Microcosm of the Land of the CavesCriteriaCultural: (v)Reference1370Inscription2014 (38th Session)Coordinates31°35′35″N 34°53′54″E / 31.59306°N 34.89833°E / 31.59306; 34.89833Location of Maresha in Israel Tel Maresha (Hebrew: תל מראשה) is the tell (archaeological mound) of the ...

Taoism

Religious and philosophical tradition This article may be in need of reorganization to comply with Wikipedia's layout guidelines. Please help by editing the article to make improvements to the overall structure. (December 2023) (Learn how and when to remove this message) TaoismThe Chinese character 道, which represents the Tao and is often translated as 'way', 'path', 'technique', or 'doctrine'Chinese nameChinese道教Hanyu PinyinDàojiào Literal meaningReligion of the WayTranscriptionsStan...

Caracalla

Roman emperor from 198 to 217 This article is about the Roman emperor. For the felid species, see Caracal. For the racehorse, see Caracalla (horse). CaracallaBust of Caracalla, Museo Nazionale Romano, 212–215 ADRoman emperorReign28 January 198 – 8 April 217 (senior from 4 February 211)PredecessorSeptimius SeverusSuccessorMacrinusCo-rulersSeptimius Severus (198–211)Geta (209–211)BornLucius Septimius Bassianus4 April 188LugdunumDied8 April 217 (aged 29)On the road between Edessa and Car...

Gylfi Þorsteinsson Gíslason

Icelandic politician (1917–2004) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Gylfi Þorsteinsson Gíslason – news · newspapers · books · scholar · JSTOR (January 2021) (Learn how and when to remove this message) This is an Icelandic name. The last name is patronymic, not a family name; this person is r...

Guayaquil

Si ce bandeau n'est plus pertinent, retirez-le. Cliquez ici pour en savoir plus. Cet article ne cite pas suffisamment ses sources (mai 2021). Si vous disposez d'ouvrages ou d'articles de référence ou si vous connaissez des sites web de qualité traitant du thème abordé ici, merci de compléter l'article en donnant les références utiles à sa vérifiabilité et en les liant à la section « Notes et références ». En pratique : Quelles sources sont attendues ? Comme...

Lower Rhine region

You can help expand this article with text translated from the corresponding article in German. Click [show] for important translation instructions. Machine translation, like DeepL or Google Translate, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Wikipedia. Do not translate text that appears unreliable or low-quality. If possibl...

Kfar Tavor

Town in northern Israel Local councilKfar Tavor כְּפַר תָּבוֹר‎كفر تافورLocal council (from 1949)Watchmen's Square in Kfar Tavor, sculptor: Asaf LifshitzKfar TavorShow map of Northeast IsraelKfar TavorShow map of IsraelCoordinates: 32°41′13″N 35°25′15″E / 32.68694°N 35.42083°E / 32.68694; 35.42083Grid position189/232 PALDistrictNorthernFounded1901Government • Head of MunicipalityOded HalperinArea • Tot...