# 概率分布

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.[1][2] It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).[3]

In Probability Theory and Statistics, a Probability Distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

--普天星相(讨论)  【审校】“ 概率分布 Probability Distribution是一个给出一个实验不同可能结果出现的概率的数学函数。”一句改为“ 概率分布 Probability Distribution是一个数学函数，它给出一个试验不同可能结果出现的概率。”
--普天星相(讨论)  【审校】“它是根据 样本空间 Sample Space 事件概率 Probabilities of Events(样本空间的子集)对随机现象的数学描述。”一句中“(样本空间的子集)”改为“（事件即样本空间的子集）”。

For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.[4]

For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.

--普天星相(讨论)  【审校】“例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。”一句改为“例如，如果使用随机变量X来表示掷硬币（“试验”）的结果，那么X的概率分布是：X = 正面的概率值为0.5，X = 反面的概率值为0.5（假设硬币是公平的）。”

A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values. For example, the sample space of a coin flip would be {heads, tails} .

A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values. For example, the sample space of a coin flip would be .

--普天星相(讨论)  【审校】“它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。”一句改为“它可能是实数的集合或向量的集合，也可能是非数值的集合。例如，抛硬币的样本空间是{正面，反面}。”

Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function.[4][5] The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

--普天星相(讨论)  【审校】“ 离散概率分布 Discrete Probability Distribution适用于一组可能的结果是离散的情况”一句中“一组可能的结果”改为“可能结果的集合”。
--普天星相(讨论)  【审校】“ 连续概率分布 Continuous Probability Distribution适用于一组可以在一个连续的范围内取值的结果的情况”一句中“一组可以在一个连续的范围内取值的结果”改为“可能结果集在连续范围内取值”。
--普天星相(讨论)  【审校】“更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。”一句改为“更复杂的试验，例如那些涉及用连续时间定义的随机过程的试验，可能需要使用更具有一般性的概率测度。”

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

--普天星相(讨论)  【审校】“而样本空间为二维或更多向量空间的分布被称为 多变量 Multivariate”一句中“二维或更多向量空间”改为“二维或更多维向量空间”。
--普天星相(讨论)  【审校】“单变量分布给出了单个随机变量取不同替代值的概率”一句中“不同替代值”改为“不同可能值”。
--普天星相(讨论)  【审校】“联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。”一句改为“多变量分布（联合概率分布）给出了一个随机向量（两个或多个随机变量组成的列表）各种取值组合的概率。”

## Introduction 简介

[[File:Dice Distribution (bar).svg|thumb|250px|right| 图1:The probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.

The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]

--普天星相(讨论)  【审校】“概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布”一句中“指定”改为“列出了”。
--普天星相(讨论)  【审校】“Pmf 允许计算事件的概率”一句中“允许”改为“可用于”。

To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function $\displaystyle{ p }$ assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

$\displaystyle{ p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2. }$

--普天星相(讨论)  【审校】“6个值中的每一个的概率为1/6”一句中“6个值”改为“1-6这六个值”。

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

--普天星相(讨论)  【审校】“那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率”一句改为“通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件（例如区间）才有正的概率”。
--普天星相(讨论)  【审校】“人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98%”一句改为“人们可能会要求介于490克至510克之间的“500克”包装出现的概率不低于98%”。

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

--普天星相(讨论)  【审校】“并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。”一句改为“并且结果落在给定区间内的概率可以通过对该区间的概率密度函数进行积分来计算。”

--普天星相(讨论)  【审校】图注中“它是概率密度曲线下面的区域”一句中“下面的区域”改为“下方的面积”。

The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity $\displaystyle{ \infty }$ to $\displaystyle{ x }$ as described by the picture to the right.[6]

The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.

--普天星相(讨论)  【审校】“累积分布函数指标描述了随机变量不大于给定值的概率”一句中，删除“指标”。
--普天星相(讨论)  【审校】“结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算”一句改为“结果落在给定区间内的概率可以通过累积分布函数在区间端点的值之间的差来计算。”
--普天星相(讨论)  【审校】“累积分布函数是从负无穷到 x 的概率密度函数下面的区域”一句中“下面的区域”改为“下方的面积”。

[[File:Standard deviation diagram.svg|right|thumb|250px| 图3:The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve. 正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]

--普天星相(讨论)  【审校】图注中“也称为高斯或钟形曲线”一句中“高斯”改为“高斯分布”。
--普天星相(讨论)  【审校】图注中“值间隔的概率对应于曲线下面积”一句中“值间隔的概率”改为“区间里的值所表示的概率”。

## Terminology 术语

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.[1]

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X > a, X < b or a union thereof.

--普天星相(讨论)  【审校】此处英文后半句应位于下文，重复出现。

### Functions for discrete variables 离散变量函数

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.

• Probability function: describes the probability distribution of a discrete random variable.

For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote

--普天星相(讨论)  【审校】此句应位于下文，重复出现。
• Relative frequency distribution: a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample i.e. sample size.

--普天星相(讨论)  【审校】本句中“多个结果”改为“结果数”。
• Discrete probability distribution function: general term to indicate the way the total probability of 1 is distributed over all various possible outcomes (i.e. over entire population) for discrete random variable.

--普天星相(讨论)  【审校】本句中“整个人群”改为“整个总体”。

--普天星相(讨论)  【审校】本句中“适用于具有有限值集”改为“取值结果为有限集合”。

### Functions for continuous variables 连续变量函数

• Probability density function (pdf): function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

--普天星相(讨论)  【审校】此句改为“在样本空间（随机变量可能取值的集合）中任意给定样本（或样本点），此样本在该函数上的取值可以被解释成给出了随机变量取值等于此样本的相对可能性 relative likelihood”。

It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as

--普天星相(讨论)  【审校】此句应位于下文，重复出现。
• Continuous probability distribution function: most often reserved for continuous random variables.

--普天星相(讨论)  【审校】此句“保留的”改为“用于”。

--普天星相(讨论)  【审校】此句“评估”改为“计算”；“取小于或等于x的值”改为“取值小于或等于x”。

### Basic terms 基本术语

• Mode: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.

--普天星相(讨论)  【审校】此句“模式”改为“众数”。
• Support: set of values that can be assumed, with non-zero probability, by the random variable.

--普天星相(讨论)  【审校】此句“支持”改为“支撑集”；“一组值”改为“值的集合”；“有时表示为R_ {X}”改为“它的支撑集有时表示为R_ {X}”。
• Tail:[7] the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form $\displaystyle{ X \gt a }$, $\displaystyle{ X \lt b }$ or a union thereof.

--普天星相(讨论)  【审校】此句前一半改为“尾部：当pmf或pdf相对较低时，靠近随机变量边界的区域。”
• Head:[7] the region where the pmf or pdf is relatively high. Usually has the form $\displaystyle{ a \lt X \lt b }$.

--普天星相(讨论)  【审校】此句“或其连续类似物”改为“或连续随机变量的类似取值。”

A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

--普天星相(讨论)  【审校】此段应出现在下文，此处重复。
• Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.

--普天星相(讨论)  【审校】此句改为“中位数：此值使得小于中位数的取值集合和大于中位数的取值集合各自的概率都不大于二分之一。”
• Variance: the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.

--普天星相(讨论)  【审校】此句改为“方差：关于均值的pmf或pdf的二阶矩；度量分布离散性的重要指标。”
• Standard deviation: the square root of the variance, and hence another measure of dispersion.

--普天星相(讨论)  【审校】此句改为“标准差：方差的平方根，因此是度量离散程度的另一指标。”
• 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.

• Skewness: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.

--普天星相(讨论)  【审校】此句“分布的第三个标准化时刻”改为“分布的三阶矩”。
• Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

--普天星相(讨论)  【审校】此句改为“峰度：pmf或pdf尾部“胖瘦”的量度。分布的四阶矩。”。

## Discrete probability distribution 离散概率分布

--普天星相(讨论)  【审校】图4图注未译，应为“离散概率分布的概率质量函数。单元集{1}, {3}, 和{7}的概率分别为0.2, 0.5, 0.3。不包含这些点的集合的概率是0。”。
--普天星相(讨论)  【审校】图5图注改为“离散概率分布的累积分布函数”。
--普天星相(讨论)  【审校】图6图注改为“连续概率分布的累积分布函数”。
--普天星相(讨论)  【审校】图7图注改为“既有连续部分又有离散部分的分布的累积分布函数”。

A discrete probability distribution is a probability distribution that can take on a countable number of values.[8] For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if $\displaystyle{ \operatorname{P}(X=n) = \tfrac{1}{2^n} }$ for n = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.

--普天星相(讨论)  【审校】“在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1”一句中“无限大”改为“可数无穷大”。
--普天星相(讨论)  【审校】含公式句中“如果$\displaystyle{ \operatorname{P}(X=n) = \tfrac{1}{2^n} }$ for n = 1, 2”改为“如果对于n = 1, 2, ...有$\displaystyle{ \operatorname{P}(X=n) = \tfrac{1}{2^n} }$”。

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.[3] Additionally, the 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。

--普天星相(讨论)  【审校】“统计建模中使用的众所周知的离散概率分布包括”中“使用的众所周知的”改为“常用的”。

When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete and that provides information about the population distribution. 当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。

### Measure theoretic formulation 测量理论公式

A measurable function $\displaystyle{ X \colon A \to B }$ between a probability space $\displaystyle{ (A, \mathcal A, P) }$ and a measurable space $\displaystyle{ (B, \mathcal B) }$ is called a discrete random variable provided that its image is a countable set. In this case measurability of $\displaystyle{ X }$ means that the pre-images of singleton sets are measurable, i.e., $\displaystyle{ X^{-1}(\{b\}) \in \mathcal A }$ for all $\displaystyle{ b \in B }$. The latter requirement induces a probability mass function $\displaystyle{ f_X \colon X(A) \to \mathbb R }$ via $\displaystyle{ f_X(b):=P(X^{-1}(\{b\})) }$. Since the pre-images of disjoint sets are disjoint,

$\displaystyle{ \sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1. }$

This recovers the definition given above.

--普天星相(讨论)  【审校】“一个可测量的函数 $\displaystyle{ X \colon A \to B }$ 在一个概率空间中 $\displaystyle{ (A, \mathcal A, P) }$ and 和一个可测量空间 $\displaystyle{ (B, \mathcal B) }$ 被叫做离散随机变量。该图像是一个可数的集合。”一句改为“在概率空间$\displaystyle{ (A, \mathcal A, P) }$ 和可测空间之间的一个可测函数 $\displaystyle{ X \colon A \to B }$称为离散随机变量，它的像是一个可数集合。”
--普天星相(讨论)  【审校】“在这种情况下$\displaystyle{ X }$的测量意味着单例集的原像是可测量的 i.e., $\displaystyle{ X^{-1}(\{b\}) \in \mathcal A }$ 对于所有的$\displaystyle{ b \in B }$.”一句改为“在这种情况下$\displaystyle{ X }$的可测量性意味着单元集的原像是可测量的，即对于所有的$\displaystyle{ b \in B }$，有 $\displaystyle{ X^{-1}(\{b\}) \in \mathcal A }$ 。”

$\displaystyle{ \sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1. }$

--普天星相(讨论)  【审校】此句中“后者需要”改为“后一个必要条件”。

### 累积分布函数 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 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.

--普天星相(讨论)  【审校】此句中“密集的实数集”改为“实数的稠密集”。

### 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 functions, 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.[9]

--普天星相(讨论)  【审校】标题“三角函数表示”改为“&delta函数表示”。
--普天星相(讨论)  【审校】“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”改为“离散概率分布通常表示为包含狄拉克&delta函数的广义概率密度函数”。

### Indicator-function representation 指标功能表示

--普天星相(讨论)  【审校】标题“指标功能表示”改为“指示函数表示”。

For a discrete random variable X, let u0, u1, ... 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

$\displaystyle{ \Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots }$
--普天星相(讨论)  【审校】标题“表示”改为“令”。

$\displaystyle{ P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1. }$

--普天星相(讨论)  【审校】标题“写入为”改为“写为”。
$\displaystyle{ X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega) }$

except on a set of probability zero, where $\displaystyle{ 1_A }$ 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

A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line.[10] They are uniquely characterized by a cumulative density function模板:Dn that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and 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。

--普天星相(讨论)  【审校】“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。”一句改为“连续概率分布是支撑集为不可数集的概率分布，如实数轴上的区间。”
--普天星相(讨论)  【审校】“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]”一句中，“支撑的每个子集”改为“支撑集每个子集”。

A random variable $\displaystyle{ X }$ has a continuous probability distribution if there is a function $\displaystyle{ f: \mathbb{R} \rightarrow [0, \infty) }$ such that for each interval $\displaystyle{ I \subset \mathbb{R} }$ the probability of $\displaystyle{ X }$ belonging to $\displaystyle{ I }$ is given by the integral of $\displaystyle{ f }$ over $\displaystyle{ I }$.[11] For example, if $\displaystyle{ I = [a, b] }$ then we would have:

$\displaystyle{ \operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx }$

$\displaystyle{ \operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx }$

--普天星相(讨论)  【审校】“例如”前一句中，将“如果”以后内容作为前半句，“对于”前加“使得”二字，“$\displaystyle{ I \subset \mathbb{R} }$”后加逗号，“$\displaystyle{ I }$的概率 $\displaystyle{ X }$”改为“随机变量 $\displaystyle{ X }$属于$\displaystyle{ I }$的概率”，“一个随机变量$\displaystyle{ X }$ 有一个连续的概率分布”改为“则称这个随机变量$\displaystyle{ X }$ 有一个连续的概率分布”并置于后半句。

In particular, the probability for $\displaystyle{ X }$ to take any single value $\displaystyle{ a }$ (that is $\displaystyle{ a \le X \le a }$) 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

$\displaystyle{ F(x) = \operatorname{P}\left[-\infty \lt X \le x\right] = \int_{-\infty}^x f(x) \, dx }$

--普天星相(讨论)  【审校】“特别是”改为“特别地”，“其累积密度函数定义为”下面应有上方那行公式。

which, by this definition, has the properties: 根据定义有以下一些性质

• $\displaystyle{ F(x) }$ is non-decreasing;
• $\displaystyle{ 0 \le F(x) \le 1 }$;
• $\displaystyle{ \lim_{x \rightarrow -\infty} F(x) = 0 }$ and $\displaystyle{ \lim_{x \rightarrow \infty} F(x) = 1 }$;
• $\displaystyle{ P(a \le X \lt b) = F(b) - F(a) }$; and
• $\displaystyle{ F(x) }$ is continuous (due to the Riemann integral properties).
--普天星相(讨论)  【审校】以上性质没有翻译。第一行应为“$\displaystyle{ F(x) }$是非减的”；第三行"and"改为“和”；第四行"and"改为“以及”；第五行应为“由于黎曼积分的性质，$\displaystyle{ F(x) }$连续”

It is also possible to think in the opposite direction, which allows more flexibility. Say $\displaystyle{ F(x) }$ is a function that satisfies all but the last of the properties above, then $\displaystyle{ F }$ represents the cumulative density function for some random variable: a discrete random variable if $\displaystyle{ F }$ is a step function, and a continuous random variable otherwise.[13] 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。

--普天星相(讨论)  【审校】“假设 f(x)是满足上述所有性质的函数”改为“假设 f(x)是满足上述除最后一条外其他所有性质的函数”，“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“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”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

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 absolutely continuous. Equivalently, it is a probability distribution on the real numbers that is absolutely continuous with respect to the Lebesgue measure. Such distributions can be represented by their probability density functions. If $\displaystyle{ X }$ is such an absolutely continuous random variable, then it has a probability density function $\displaystyle{ f(x) }$, and its probability of falling into a Lebesgue-measurable set $\displaystyle{ A \subset \mathbb{R} }$ is:

$\displaystyle{ \operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu }$

where $\displaystyle{ \mu }$ is the Lebesgue measure.

$\displaystyle{ \operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu }$

--普天星相(讨论)  【审校】“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”中“实线”改为“实数轴”。“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“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”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are continuous, rather than absolutely continuous. These distributions are the ones $\displaystyle{ \mu }$ such that $\displaystyle{ \mu\{x\}\,=\,0 }$ for all $\displaystyle{ \,x }$. This definition includes the (absolutely) continuous distributions defined above, but it also includes singular distributions, 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.

--普天星相(讨论)  【审校】“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。”改为“这种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，同时也没有密度。”

## Kolmogorov definition 柯尔莫哥洛夫的定义

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function $\displaystyle{ X }$ from a probability space $\displaystyle{ (\Omega, \mathcal{F}, \mathbb{P}) }$ to a measurable space $\displaystyle{ (\mathcal{X},\mathcal{A}) }$. Given that probabilities of events of the form $\displaystyle{ \{\omega\in\Omega\mid X(\omega)\in A\} }$ satisfy Kolmogorov's probability axioms, the probability distribution of X is the pushforward measure $\displaystyle{ X_*\mathbb{P} }$ of $\displaystyle{ X }$ , which is a probability measure on $\displaystyle{ (\mathcal{X},\mathcal{A}) }$ satisfying $\displaystyle{ X_*\mathbb{P} = \mathbb{P}X^{-1} }$.[14][15][16]

--普天星相(讨论)  【审校】“在概率论的度量理论形式化中”一句中“度量理论”改为“测度论”。
--普天星相(讨论)  【审校】“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）”一句改为“随机变量定义为从概率空间$\displaystyle{ (\Omega, \mathcal{F}, \mathbb{P}) }$到测度空间$\displaystyle{ (\mathcal{X},\mathcal{A}) }$的测度函数$\displaystyle{ (\mathcal{X},\mathcal{A}) }$。”
--普天星相(讨论)  【审校】“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”一句改为“假设这种形式的事件概率$\displaystyle{ \{\omega\in\Omega\mid X(\omega)\in A\} }$满足满足柯尔莫哥洛夫的概率公理，则X的概率分布为X的pushforward measure$\displaystyle{ X_*\mathbb{P} }$”。

## Random number generation 随机数生成

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.[17]

--普天星相(讨论)  【审校】“半开间隔”改为“半开区间”。

For example, suppose $\displaystyle{ U }$ has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some $\displaystyle{ 0 \lt p \lt 1 }$, we define

$\displaystyle{ {\displaystyle X ={\begin{cases}1,&{\mbox{if }}U\lt p\\0,&{\mbox{if }}U\geq p\end{cases}}} }$

so that

$\displaystyle{ \textrm{P}(X=1) = \textrm{P}(U\lt p) = p, \textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p. }$

$\displaystyle{ {\displaystyle X ={\begin{cases}1,&{\mbox{if }}U\lt p\\0,&{\mbox{if }}U\geq p\end{cases}}} }$

$\displaystyle{ \textrm{P}(X=1) = \textrm{P}(U\lt p) = p, \textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p. }$

--普天星相(讨论)  【审校】“因此”改为“使得”。

This random variable X has a Bernoulli distribution with parameter $\displaystyle{ p }$.[17] 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。

--普天星相(讨论)  【审校】“该随机变量X具有参数的伯努利分布p”改为“该随机变量X具有以p为参数的伯努利分布”。

For a distribution function $\displaystyle{ F }$ of a continuous random variable, a continuous random variable must be constructed. $\displaystyle{ F^{inv} }$, an inverse function of $\displaystyle{ F }$, relates to the uniform variable $\displaystyle{ U }$:

$\displaystyle{ {U\leq F(x)} = {F^{inv}(U)\leq x}. }$

F或连续随机变量的分布函数F，必须构造连续随机变量。 $\displaystyle{ F^{inv} }$，F的反函数，涉及均匀变量U： $\displaystyle{ {U\leq F(x)} = {F^{inv}(U)\leq x}. }$

--普天星相(讨论)  【审校】“F或连续随机变量的分布函数F”中“F或”改为“对于”。
--普天星相(讨论)  【审校】“F的反函数，涉及均匀变量U”改为“F反函数$\displaystyle{ F^{inv} }$与均匀变量U有以下关系：”。

For example, suppose a random variable that has an exponential distribution $\displaystyle{ F(x) = 1 - e^{-\lambda x} }$ must be constructed. 例如，假设必须构造一个具有指数分布$\displaystyle{ F(x) = 1 - e^{-\lambda x} }$ 的随机变量。

\displaystyle{ \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} }

--fairywang(讨论)  【审校】"并且如果 $\displaystyle{ U }$ 有一个$\displaystyle{ U(0,1) }$ 分布, 然后"改为“如果 $\displaystyle{ U }$ 服从$\displaystyle{ (0,1) }$ 分布, 且”

A frequent problem in statistical simulations (the Monte Carlo method) is the generation of 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

--fairywang(讨论)  【审校】“共同概率分布及其应用”改为“常见概率分布及其应用”

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 properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

--fairywang(讨论)  【审校】“简单数字通常很难表述一个量量”改为“简单数字通常不足以表述一个量”

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.) 以下列出了一些最常见的概率分布，按与之相关的过程类型进行分组。更完整的内容请参见概率分布列表，该列表按研究对象的性质（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是单峰的。也就是说，假设值聚集在单个值周围。实际上，实际观测量可能会聚集在多个值附近，可以使用混合分布对这种量进行建模。

### Linear growth (e.g. errors, offsets) 线性增长

--fairywang(讨论)  【审校】“线性增长”改为“线性增长（例如，错误，偏差等）”
• Normal distribution (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution

### Exponential growth (e.g. prices, incomes, populations) 指数增长

--fairywang(讨论)  【审校】“指数增长”改为“指数增长（例如，价格，收入，人口）”

### Bernoulli trials (yes/no events, with a given probability) 伯努利试验（给定的概率的是或否事件

--fairywang(讨论)  【审校】“给定的概率的是或否事件”改为“给定了概率的是或否事件”
• Basic distributions 基本分布:

• 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

• Related to sampling schemes over a finite population 与有限人口抽样方案有关:

### 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

### 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.

• 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.

### 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）

F-分布 F-Distribution，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

Beta分布 Beta Distribution，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

Dirichlet分布 Dirichlet Distribution，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分布 Wishart Distribution，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

--fairywang(讨论)  【审校】“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。”改为“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，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 $\displaystyle{ P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 }$, probability that the particle's position x will be in the interval axb in dimension one, and a similar triple integral in dimension three. This is a key principle of quantum mechanics.[19]

• 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.[20]

Copula（统计数据）

Riemann–Stieltjes积分＃在概率论中的应用

### Lists 清单

--fairywang(讨论)  【审校】“统计学话题的清淡”改为“统计学话题的清单”

## References

### Citations

1. Everitt, Brian. (2006). The Cambridge dictionary of statistics (3rd ed.). Cambridge, UK: Cambridge University Press. ISBN 978-0-511-24688-3. OCLC 161828328.
2. Ash, Robert B. (2008). Basic probability theory (Dover ed.). Mineola, N.Y.: Dover Publications. pp. 66–69. ISBN 978-0-486-46628-6. OCLC 190785258.
3. Evans, Michael (Michael John) (2010). Probability and statistics : the science of uncertainty. Rosenthal, Jeffrey S. (Jeffrey Seth) (2nd ed.). New York: W.H. Freeman and Co. pp. 38. ISBN 978-1-4292-2462-8. OCLC 473463742.
4. Ross, Sheldon M. (2010). A first course in probability. Pearson.
5. DeGroot, Morris H.; Schervish, Mark J. (2002). Probability and Statistics. Addison-Wesley.
6. A modern introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005. ISBN 978-1-85233-896-1. OCLC 262680588.
7. More information and examples can be found in the articles Heavy-tailed distribution, Long-tailed distribution, fat-tailed distribution
8. Erhan, Çınlar (2011). Probability and stochastics. New York: Springer. pp. 51. ISBN 9780387878591. OCLC 710149819.
9. Khuri, André I. (March 2004). "Applications of Dirac's delta function in statistics". International Journal of Mathematical Education in Science and Technology (in English). 35 (2): 185–195. doi:10.1080/00207390310001638313. ISSN 0020-739X.
10. Sheldon M. Ross (2010). Introduction to probability models. Elsevier.
11. Chapter 3.2 of 模板:Harvp
12. Chapter 3.2 of 模板:Harvp
13. See Theorem 2.1 of 模板:Harvp, or Lebesgue's decomposition theorem. The section #Delta-function_representation may also be of interest.
14. W., Stroock, Daniel (1999). Probability theory : an analytic view (Rev. ed.). Cambridge [England]: Cambridge University Press. pp. 11. ISBN 978-0521663496. OCLC 43953136.
15. Kolmogorov, Andrey (1950). Foundations of the theory of probability. New York, USA: Chelsea Publishing Company. pp. 21–24.
16. Joyce, David (2014). "Axioms of Probability" (PDF). Clark University. Retrieved December 5, 2019.
17. Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005), "Why probability and statistics?", A Modern Introduction to Probability and Statistics, Springer London, pp. 1–11, doi:10.1007/1-84628-168-7_1, ISBN 978-1-85233-896-1
18. Bishop, Christopher M. (2006). Pattern recognition and machine learning. New York: Springer. ISBN 0-387-31073-8. OCLC 71008143.
19. Chang, Raymond.. Physical chemistry for the chemical sciences. Thoman, John W., Jr., 1960-. [Mill Valley, California]. pp. 403–406. ISBN 978-1-68015-835-9. OCLC 927509011.
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