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Baris ka-3: :<math>F(x) = \operatorname{P}(X\leq x),</math> numana sisi beulah katuhu ngagambarkeun [[probabilitas]] numana variabel acak ''X'' dicokot tina nilai nu kurang tina atawa sarua jeung ''x''. Probabilitas ''X'' aya dina [[interval (mathematics)\|interval]] (''a'', ''b''<nowiki>]</nowiki> nyaeta ''F''(''b'') − ''F''(''a'') lamun ''a'' < ''b''. It is conventional to use a capital ''F'' for a cumulative distribution function, in contrast to the lower-case ''f'' used for [[probability density function]]s and [[probability mass function]]s. ~~the [[probability]] that the variable ''X'' takes on a value less than or equal to ''x''.~~ The probability that ''X'' lies in the [[interval (mathematics)\|interval]] (''a'', ''b'') is therefore ''F''(''b'') − ''F''(''a'') if ''a'' ≤ ''b''. It is conventional to use a capital ''F'' for a cumulative distribution function, in contrast to the lower-case ''f'' used for [[probability density function]]s and probability mass functions. The CDF of X can be defined in terms of the [[probability density function]] ''f'' as follows: Note that in the definition above, the "less or equal" sign, '≤' could be replaced with "strictly less" '<'. This would yield a different function, but either of the two functions can be readily derived from the other. One could even use "greater" sign there (changing cdf properties even more). The only thing to remember is to stick to either definition as mixing them will lead to incorrect results. In English-speaking countries the convention that uses the weak inequality (≤) rather than the strict inequality (<) is nearly always used. :<math>F(x) = \int_{-\infty}^x f(t)\,dt</math> ~~== Conto ==~~ Note that in the definition above, the "less or equal" sign, '≤' is a convention, but it is ~~Salaku conto, anggap ''X'' kasebar seragam dina [[unit interval]] [0, 1].~~ an important and universally used one. The proper use of tables of the Binomial and Poisson ~~Mangka cdf nyaeta~~ distributions depend upon this convention. Moreover, important formulas like Levy's inversion formula for the characteristic function also rely on the "less or equal" formulation. == Properties == ~~:''F''(''x'') = 0, lamun ''x'' < 0;~~ [[Image:Discrete probability distribution illustration.png\|right\|thumb\|From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part.]] ~~:''F''(''x'') = ''x'', lamun 0 ≤ ''x'' ≤ 1;~~ Every cumulative distribution function ''F'' is (not necessarily strictly) [[monotone increasing]] and [[right-continuous]]. Furthermore, we have ~~:''F''(''x'') = 1, lamun ''x'' > 1.~~ :<math>\lim_{x\to -\infty}F(x)=0, \quad \lim_{x\to +\infty}F(x)=1.</math> Every function with these four properties is a cdf. The properties imply that all CDFs are [[càdlàg]] functions. ~~Contona sejenna, anggap ''X'' ngan nilai 0 jeung 1, proababiliti sarua.~~ ~~Mangka cdf nyaeta~~ If ''X'' is a [[discrete random variable]], then it attains values ''x''<sub>1</sub>, ''x''<sub>2</sub>, ... with probability ''p''<sub>i</sub> = P(''x''<sub>i</sub>), and the cdf of ''X'' will be discontinuous at the points ''x''<sub>''i''</sub> and constant in between: ~~:''F''(''x'') = 0, lamun ''x'' < 0;~~ ~~:''F''(''x'') = 1/2, lamun 0 ≤ ''x'' < 1;~~ ~~:''F''(''x'') = 1, lamun ''x'' ≥ 1.~~ :<math>F(x) = \operatorname{P}(X\leq x) = \sum_{x_i \leq x} \operatorname{P}(X = x_i) = \sum_{x_i \leq x} p(x_i)</math> ~~== Sipat ==~~ If the CDF ''F'' of ''X'' is [[continuous function\|continuous]], then ''X'' is a [[continuous random variable]]; if furthermore ''F'' is [[absolute continuity\|absolutely continuous]], then there exists a [[Lebesgue integral\|Lebesgue-integrable]] function ''f''(''x'') such that Every cumulative distribution function ''F'' is [[monotone increasing]] and [[continuous]] from the right. Furthermore, we have [[limit (mathematics)\|lim]]<sub>''x'' → −∞</sub> ''F''(''x'') = 0 and lim<sub>''x'' → +∞</sub> ''F''(''x'') = 1. Every function with these four properties is a cdf. :<math>F(b)-F(a) = \operatorname{P}(a\leq X\leq b) = \int_a^b f(x)\,dx</math> If ''X'' is a [[discrete random variable]], then it attains values ''x''<sub>1</sub>, ''x''<sub>2</sub>, ... with probability ''p''<sub>1</sub>, ''p''<sub>2</sub> etc., and the cdf of ''X'' will be discontinuous at the points ''x''<sub>''i''</sub> and constant in between. for all real numbers ''a'' and ''b''. (The first of the two equalities displayed above would not be correct in general if we had not said that the distribution is continuous. Continuity of the distribution implies that P(''X'' = ''a'') = P(''X'' = ''b'') = 0, so the difference between "<" and "≤" ceases to be important in this context.) The function ''f'' is equal to the [[derivative]] of ''F'' [[almost everywhere]], and it is called the [[probability density function]] of the distribution of ''X''. If the cdf ''F'' of ''X'' is [[continuous]], then ''X'' is a [[continuous random variable]]; if furthermore ''F'' is [[absolute continuity\|absolutely continuous]], then there exists a [[Lebesgue integral\|Lebesgue-integrable]] function ''f''(''x'') such that ===Point probability=== ~~:<math>F(b)-F(a) = \operatorname{P}(a\leq X\leq b) = \int_a^b f(x)\,dx</math>~~ The "point probability" that ''X'' is exactly ''b'' can be found as :<math>\operatorname{P}(X=b) = F(b) - \lim_{x \to b^{-}} F(x)</math> for all real numbers ''a'' and ''b''. (The first of the two equalities displayed above would not be correct in general if we had not said that the distribution is continuous. Continuity of the distribution implies that P(''X'' = ''a'') = P(''X'' = ''b'') = 0, so the difference between "<math><</math>" and "<math>\leq</math>" ceases to be important in this context.) The function ''f'' is equal to the [[derivative]] of ''F'' [[almost everywhere]], and it is called the [[probability density function]] of the distribution of ''X''. ==Kolmogorov-Smirnov and Kuiper's tests== The [[Kolmogorov-Smirnov test]] is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related [[Kuiper's test]] (pronounced in Dutch the way an Cowper might be pronounced in English) is useful if the domain of the distribution is cyclic as in day of the week. For instance we might use Kuiper's test to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month. The [[Kolmogorov-Smirnov test]] is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related [[Kuiper's test]] (pronounced {{IPA\|/kœypəʁ/}}) is useful if the domain of the distribution is cyclic as in day of the week. For instance we might use Kuiper's test to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month. ==Complementary cumulative distribution function==<!-- This section is linked from [[Power law]] --> Sometimes, it is useful to study the opposite question and ask how often the random variable is ''above'' a particular level. This is called the '''complementary cumulative distribution function''' ('''ccdf'''), defined as :<math>F_c(x) = \operatorname{P}(X > x) = 1 - F(x)</math>. In survival analysis, <math>F_c(x)</math> is called the '''survival function''' and denoted <math> S(x) </math>. == Examples == As an example, suppose ''X'' is uniformly distributed on the unit interval [0, 1]. Then the CDF of X is given by :<math>F(x) = \begin{cases} 0 &:\ x < 0\\ x &:\ 0 \le x \le 1\\ 1 &:\ 1 < x \end{cases}</math> Take another example, suppose ''X'' takes only the discrete values 0 and 1, with equal probability. Then the CDF of X is given by :<math>F(x) = \begin{cases} 0 &:\ x < 0\\ 1/2 &:\ 0 \le x < 1\\ 1 &:\ 1 \le x \end{cases}</math> ==Inverse== If the cdf ''F'' is strictly increasing and continuous then <math> F^{-1}( y ), y \in [0,1] </math> is the unique real number <math> x </math> such that <math> F(x) = y </math>. Unfortunately, the distribution does not, in general, have an inverse. One may define, for <math> y \in [0,1] </math>, :<math> F^{-1}( y ) = \inf_{r \in \mathbb{R}} \{ F( r ) > y \} </math>. Example 1: The median is <math>F^{-1}( 0.5 )</math>. Example 2: Put <math> \tau = F^{-1}( 0.95 ) </math>. Then we call <math> \tau </math> the 95th percentile. The inverse of the cdf is called the [[quantile function]]. ==Tempo oge== * [[Descriptive statistics]] * [[Probability distribution]] * [[Empirical distribution function]] * [[Cumulative frequency]] distributions * [[Q-Q plot]] * [[Ogive#Statistics\|Ogive]] * [[Quantile function]] == Rujukan == ~~[[Statistik deskriptif]], [[Probability distribution]]~~ <references/> [[Category:Probability theory]] == Tumbu kaluar == *[http://stattrek.com/Lesson2/DiscreteContinuous.aspx?Tutorial=Stat An introduction to probability distributions] [[Kategori:Statistika]] [[da:Fordelingsfunktion]] [[de:~~Empirische~~ Verteilungsfunktion]] ~~[[en:Cumulative distribution function]]~~ [[fr:Fonction de répartition]] ~~[[hu:Eloszlásfüggvény]]~~ [[it:Funzione di ripartizione]] [[hu:Eloszlásfüggvény]] [[pl:Dystrybuanta]] [[pt:Função distribuição acumulada]] [[ru:Функция распределения]] [[su:Fungsi sebaran kumulatif]] [[sv:Kumulativ fördelningsfunktion]] [[vi:Hàm phân bố tích lũy]] [[zh:累积分布函数]]

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