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第239行： − − It is also possible to think in the opposite direction, which allows more flexibility. Say F(x) is a function that satisfies all but the last of the properties above, then F represents the cumulative density function for some random variable: a discrete random variable if F is a step function, and a continuous random variable otherwise. This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the Cantor distribution. − − 也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。 − 第263行：
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第262行： − In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function X from a probability space (\Omega, \mathcal{F}, \mathbb{P}) to a measurable space (\mathcal{X},\mathcal{A}). Given that probabilities of events of the form \{\omega\in\Omega\mid X(\omega)\in A\} satisfy Kolmogorov's probability axioms, the probability distribution of X is the pushforward measure X_*\mathbb{P} of X , which is a probability measure on (\mathcal{X},\mathcal{A}) satisfying X_*\mathbb{P} = \mathbb{P}X^{-1}.+ − 在概率论的度量理论形式化中，一个随机变量被定义为一个可测函数 x，从一个概率空间(Omega，mathcal { f } ，mathbb { p })到一个可测空间(mathcal { x } ，mathcal { a })。考虑到 a }中 Omega 中 x (Omega)中 ω 的事件概率满足 Kolmogorov 的概率公理，x 的概率分布是 x 的 x * mathbb { p }的前推测度，它是关于(cal { x } ，mathcal { a })满足 x * mathbb { p } = mathbb { p } x ^ {-1}的机率量测。 + + − ===Delta-function representation===+ − Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.<ref>{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}</ref>+ + + + + + + − ===Indicator-function representation===+ − Most algorithms are based on a pseudorandom number generator that produces numbers X that are uniformly distributed in the half-open interval [0,1). These random variates X are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.+ − 大多数算法都是基于一个伪随机数生成器，它产生的数字 x 在半开区间内均匀分布。这些随机变量 x 然后通过一些算法转换，创建一个新的具有所需概率分布的随机变量。有了这种均匀的伪随机性，任何随机变量都可以被实现。+ + − For a discrete random variable ''X'', let ''u''<sub>0</sub>, ''u''<sub>1</sub>, ... be the values it can take with non-zero probability. Denote+ + − For example, suppose U has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some 0 < p < 1, we define − 例如，假设 u 在0和1之间有一个均匀分布。为了构造一个0 < p < 1的随机贝努利变量，我们定义了+ + + + − + + + − + − { displaystyle x = { begin { cases }1，& { mbox { if } u < p 0，& { mbox { if } u geq p end { cases }}+ − These are [[disjoint set]]s, and for such sets+ − </math>+ + − 数学+ + + − + − so that+ − 所以+ + + + − It follows that the probability that ''X'' takes any value except for ''u''<sub>0</sub>, ''u''<sub>1</sub>, ... is zero, and thus one can write ''X'' as − <math>\textrm{P}(X=1) = \textrm{P}(U<p) = p, − + + + − \textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>+ − 文本{ p }(x = 0) = 文本{ p }(u geq p) = 1-p. </math > + − :<math>X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)</math>+ + + − This random variable X has a Bernoulli distribution with parameter p.+ − 这个随机变量 x 有一个带参数 p 的伯努利分布。+ − except on a set of probability zero, where <math>1_A</math> is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables. + − + − {{See also|Probability density function}} + + + − A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.<ref>{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}</ref> They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].+ + + + − A random variable <math>X</math> has a continuous probability distribution if there is a function <math>f: \mathbb{R} \rightarrow [0, \infty)</math> such that for each interval <math>I \subset \mathbb{R}</math> the probability of <math>X</math> belonging to <math>I</math> is given by the integral of <math>f</math> over <math>I</math>.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> For example, if <math>I = [a, b]</math> then we would have:+ − :<math>+ + − \operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx+ + − In particular, the probability for <math>X</math> to take any single value <math>a</math> (that is <math>a \le X \le a</math>) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as+ + + + + + + + + + + + + + + + + + + + + + + + + + − :<math>+ + − F(x) = \operatorname{P}\left[-\infty < X \le x\right] = \int_{-\infty}^x f(x) \, dx+ − </math>+ − which, by this definition, has the properties:+ + − <ul>+ + − <li style="margin: 0.7rem 0;"><math>F(x)</math> is non-decreasing;</li>+ − <li style="margin: 0.7rem 0;"><math>0 \le F(x) \le 1</math>;</li>+ + − <li style="margin: 0.7rem 0;"><math>\lim_{x \rightarrow -\infty} F(x) = 0</math> and <math>\lim_{x \rightarrow \infty} F(x) = 1</math>;</li>+ − <li style="margin: 0.7rem 0;"><math>P(a \le X < b) = F(b) - F(a)</math>; and</li>+ + − <li style="margin: 0.7rem 0;"><math>F(x)</math> is continuous (due to the [[Riemann integral]] properties).</li>+ + − </ul>+ − It is also possible to think in the opposite direction, which allows more flexibility. Say <math>F(x)</math> is a function that satisfies all but the last of the properties above, then <math>F</math> represents the cumulative density function for some random variable: a discrete random variable if <math>F</math> is a step function, and a continuous random variable otherwise.<ref>See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.</ref> This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].+ + + + + − It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If <math>X</math> is such an absolutely continuous random variable, then it has a [[probability density function]] <math>f(x)</math>, and its probability of falling into a Lebesgue-measurable set <math>A \subset \mathbb{R}</math> is:+ + + − : <math>+ + − \operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu+ + − </math>+ + − where <math>\mu</math> is the Lebesgue measure.+ + + + + − Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones <math>\mu</math> such that <math>\mu\{x\}\,=\,0</math> for all <math>\,x</math>. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].+ + + + + − == [[Andrey Kolmogorov|Kolmogorov]] definition ==+ + − {{Main|Probability space|Probability measure}}+ + + + + − In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] <math>X</math> from a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> to a [[measurable space]] <math>(\mathcal{X},\mathcal{A})</math>. Given that probabilities of events of the form <math>\{\omega\in\Omega\mid X(\omega)\in A\}</math> satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] <math>X_*\mathbb{P}</math> of <math>X</math> , which is a [[probability measure]] on <math>(\mathcal{X},\mathcal{A})</math> satisfying <math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>.<ref>{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}</ref><ref>{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}</ref><ref>{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}</ref>+ + + + + − ==Random number generation==+ + − {{Main|Pseudo-random number sampling}}+ + + + + − Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.<ref name=":0">{{Citation|last=Dekking|first=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}</ref>+ + + + + + − For example, suppose <math>U</math> has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some <math>0 < p < 1</math>, we define+ + + + + + − <math>{\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}+ − </math>+ + − Category:Mathematical and quantitative methods (economics)+ + − 类别: 数学和定量方法(经济学)+ + + + + + + + + + + + + + + + + + + + + + − so that+ − it:Variabile casuale#Distribuzione di probabilità+ + + + + − 它: Variabile casuale # distributione di 盖率 à+ + + + + + − <noinclude>+ + + − <small>This page was moved from [[wikipedia:en:Probability distribution]]. Its edit history can be viewed at [[概率分布/edithistory]]</small></noinclude>+ + + + + − + + + +
无编辑摘要
*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.
*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。
* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.
* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.
标准偏差：方差的平方根，因此是色散的另一种度量。
标准偏差：方差的平方根，因此是色散的另一种度量。
* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.
* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]
[[File:Discrete probability distribution.svg|right|thumb|
[[File:Discrete probability distribution.svg|right|thumb|
Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].<ref name=":1" /> Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.
Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].<ref name=":1" /> Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。
This recovers the definition given above.
This recovers the definition given above.
一个可测量的函数 <math>X \colon A \to B </math> 在一个概率空间中 <math>(A, \mathcal A, P)</math> and 和一个可测量空间 <math>(B, \mathcal B) </math> 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下<math>X</math>的测量意味着单例集的原像是可测量的 i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> 对于所有的<math>b \in B</math>.
后者需要包括概率质量函数 <math>f_X \colon X(A) \to \mathbb R</math> via <math> f_X(b):=P(X^{-1}(\{b\}))</math>. 由于不相交集的原像不相交
:<math>\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.</math>
这包含了上面所提到的定义
===Cumulative distribution function===
===Cumulative distribution function===
Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.
Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.
与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。
===Delta-function representation 三角函数表示===
Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.<ref>{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}</ref>
因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。
===Indicator-function representation 指标功能表示===
For a discrete random variable ''X'', let ''u''<sub>0</sub>, ''u''<sub>1</sub>, ... be the values it can take with non-zero probability. Denote
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示
:<math>\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>
这些是不相交的集合，对于这样的集合：
:<math>P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.</math>
因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为
:<math>X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)</math>
except on a set of probability zero, where <math>1_A</math> is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。
==Continuous probability distribution 连续概率分布==
{{See also|Probability density function}}
A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.<ref>{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}</ref> They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。
A random variable <math>X</math> has a continuous probability distribution if there is a function <math>f: \mathbb{R} \rightarrow [0, \infty)</math> such that for each interval <math>I \subset \mathbb{R}</math> the probability of <math>X</math> belonging to <math>I</math> is given by the integral of <math>f</math> over <math>I</math>.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> For example, if <math>I = [a, b]</math> then we would have:
:<math>\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>
:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>
一个随机变量<math>X</math> 有一个连续的概率分布，如果这有一个函数<math>f: \mathbb{R} \rightarrow [0, \infty)</math> 对于每一个区间<math>I \subset \mathbb{R}</math> <math>I</math>的概率 <math>X</math>是<math>f</math> 在 <math>I</math>上的积分.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> 例如, 如果<math>I = [a, b]</math> 可得到:
:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>
<math>{\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
In particular, the probability for <math>X</math> to take any single value <math>a</math> (that is <math>a \le X \le a</math>) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as
:<math>F(x) = \operatorname{P}\left[-\infty < X \le x\right] = \int_{-\infty}^x f(x) \, dx</math>
特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为
which, by this definition, has the properties:
根据定义有以下一些性质
<ul>
<li style="margin: 0.7rem 0;"><math>F(x)</math> is non-decreasing;</li>
<li style="margin: 0.7rem 0;"><math>0 \le F(x) \le 1</math>;</li>
:<math>P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.</math>
<li style="margin: 0.7rem 0;"><math>\lim_{x \rightarrow -\infty} F(x) = 0</math> and <math>\lim_{x \rightarrow \infty} F(x) = 1</math>;</li>
<li style="margin: 0.7rem 0;"><math>P(a \le X < b) = F(b) - F(a)</math>; and</li>
<li style="margin: 0.7rem 0;"><math>F(x)</math> is continuous (due to the [[Riemann integral]] properties).</li>
</ul>
It is also possible to think in the opposite direction, which allows more flexibility. Say <math>F(x)</math> is a function that satisfies all but the last of the properties above, then <math>F</math> represents the cumulative density function for some random variable: a discrete random variable if <math>F</math> is a step function, and a continuous random variable otherwise.<ref>See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.</ref> This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。
< math > textrm { p }(x = 1) = textrm { p }(u < p) = p,
It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If <math>X</math> is such an absolutely continuous random variable, then it has a [[probability density function]] <math>f(x)</math>, and its probability of falling into a Lebesgue-measurable set <math>A \subset \mathbb{R}</math> is:
:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>
where <math>\mu</math> is the Lebesgue measure.
对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：
:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>
这里，mu是Lebesgue度量
Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones <math>\mu</math> such that <math>\mu\{x\}\,=\,0</math> for all <math>\,x</math>. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].
关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。
== [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义==
{{Main|Probability space|Probability measure}}
In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] <math>X</math> from a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> to a [[measurable space]] <math>(\mathcal{X},\mathcal{A})</math>. Given that probabilities of events of the form <math>\{\omega\in\Omega\mid X(\omega)\in A\}</math> satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] <math>X_*\mathbb{P}</math> of <math>X</math> , which is a [[probability measure]] on <math>(\mathcal{X},\mathcal{A})</math> satisfying <math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>.<ref>{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}</ref><ref>{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}</ref><ref>{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}</ref>
==Continuous probability distribution==
在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足<math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>的概率量度
==Random number generation 随机数生成==
{{Main|Pseudo-random number sampling}}
Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.<ref name=":0">{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}</ref>
大多数算法基于伪随机数生成器，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。
For example, suppose <math>U</math> has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some <math>0 < p < 1</math>, we define
<math>{\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>
so that
<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>
例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 <p <1我们定义
<math>{\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>
</math>
因此
<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>
This random variable X has a Bernoulli distribution with parameter <math>p</math>.<ref name=":0"/> Note that this is a transformation of discrete random variable.
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。
For a distribution function <math>F</math> of a continuous random variable, a continuous random variable must be constructed. <math>F^{inv}</math>, an inverse function of <math>F</math>, relates to the uniform variable <math>U</math>:
<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>
F或连续随机变量的分布函数F，必须构造连续随机变量。 <math>F^{inv}</math>，F的反函数，涉及均匀变量U：
<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>
For example, suppose a random variable that has an exponential distribution <math>F(x) = 1 - e^{-\lambda x}</math> must be constructed.
例如，假设必须构造一个具有指数分布<math>F(x) = 1 - e^{-\lambda x}</math> 的随机变量。
<math>\begin{align}
F(x) = u &\Leftrightarrow 1-e^{-\lambda x} = u \\ &\Leftrightarrow e^{-\lambda x } = 1-u \\&\Leftrightarrow -\lambda x = \ln(1-u) \\ &\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)
\end{align}</math>
所以<math>F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)</math> 并且如果 <math>U</math> 有一个<math>U(0,1)</math> 分布, 然后随机变量 <math>X</math> 被定义为 <math>X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)</math>. 这里有一个指数分布 <math>\lambda</math>.<ref name=":0" />
A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。
== Common probability distributions and their applications 共同概率分布及其应用==
{{Main list|List of probability distributions}}
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.
概率分布的概念及其描述的随机变量是概率论和统计学科学的数学基础。人口中几乎可以测量的任何值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量均存在一定的固有误差；在物理学中，从气体的动力学特性到基本粒子的量子力学描述，很多过程都用概率论来描述。由于这些以及许多其他原因，简单数字通常不足以描述数量，而概率分布通常更合适。
The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)
以下是一些最常见的概率分布列表，按与之相关的过程类型进行分组。有关更完整的列表，请参见概率分布列表，该列表按要考虑的结果的性质（离散，连续，多元等）进行分组。
All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].
下面所有的单变量分布都达到了峰值。也就是说，假设值聚集在单个点周围。实际上，实际观察到的量可能会聚集在多个值附近。可以使用混合物分布对此类数量进行建模。
=== Linear growth (e.g. errors, offsets) 线性增长===
* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution
正态分布（高斯分布），对于单个这样的数量；最常用的连续分布
=== Exponential growth (e.g. prices, incomes, populations) 指数增长===
* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed
对数正态分布，对于单个此类数量的对数正态分布
* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution
帕累托分布，对于单个这样的数量，其对数呈指数分布；原型幂律分布
=== Uniformly distributed quantities 数量均匀分布===
* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)
离散均匀分布，用于有限的一组值（例如，公平死亡的结果）
* [[Continuous uniform distribution]], for continuously distributed values
连续均匀分布，用于连续分布的值
=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（是/否事件，具有给定的概率）===
* Basic distributions 基本分布:
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
伯努利分布，用于单个伯努利试验的结果（例如成功/失败，是/否）
** [[Binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences
二项式分布，对于给定固定总数的独立“出现次数”（例如，成功，赞成票等）的数量
** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
负二项分布，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数
** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]
几何分布，用于二项式观测，但是关注的数量是首次成功之前的失败数量；负二项式分布的特殊情况
* Related to sampling schemes over a finite population 与有限人口抽样方案有关:
** [[Hypergeometric distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]
超几何分布，对于“肯定出现”的数量（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换
** [[Beta-binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the "opposite" of [[sampling without replacement]])
贝塔二项式分布，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换） ）
=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===
* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]
针对单个分类结果的分类分布（例如，调查中的是/否/也许）；伯努利分布的一般化
* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]
给定总结果的固定数量，针对每种类别结果的数量的多项式分布；二项式分布的一般化
* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]
多元超几何分布，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化
=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===
* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time
泊松分布，用于给定时间段内泊松型事件的发生次数
* [[Exponential distribution]], for the time before the next Poisson-type event occurs
指数分布，在下一次泊松型事件发生之前的时间
* [[Gamma distribution]], for the time before the next k Poisson-type events occur
伽马分布，在接下来的k个泊松型事件发生之前的时间
=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===
* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
瑞利分布，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。
* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.
莱斯分布，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。
=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===
* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])
卡方分布，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）
* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])
学生t分布，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）
* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])
F-分布，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论
=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===
{{Main|Conjugate prior}}
* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]
Beta分布，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭
* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。
* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]
Dirichlet分布，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化
*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]<ref>{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}</ref>
Wishart分布，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化
=== Some specialized applications of probability distributions 概率分布的一些特殊应用===
* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。
* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by <math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.<ref>{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}</ref>
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为<math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。
* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.<ref>{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}</ref>
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。
* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.<ref>{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}</ref>
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。
==See also 另请参见==
{{Portal|Mathematics}}
*[[Copula (statistics)]]
Copula（统计数据）
* [[Empirical probability]]
经验概率
* [[Histogram]]
直方图
* [[Likelihood function]]
似然函数
* [[Kirkwood approximation]]
柯克伍德近似
* [[Moment-generating function]]
瞬时产生功能
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]
Riemann–Stieltjes积分＃在概率论中的应用
*[[Pairwise independence]]
成对独立
=== Lists 清单===
* [[List of probability distributions]]
概率分布的清单
* [[List of statistical topics]]
统计学话题的清淡
=== Probability distributions 概率分布 ===
* [[Conditional probability distribution]]
条件概率分布
* [[Joint probability distribution]]
联合概率分布
* [[Quasiprobability distribution]]
拟概率分布
== References ==
=== Citations ===
{{Reflist}}
=== Sources ===
{{refbegin|}}
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}
{{refend}}
[[Category:待整理页面]]
==External links==
{{commons|Probability distribution|Probability distribution}}
*{{springer|title=Probability distribution|id=p/p074900}}
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions], Gavin E. Crooks.