概率分布

本词条由Ryan初步翻译

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).

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).

在概率论 Probability Theory和统计学 Statistics中， 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。它是根据 样本空间 Sample Space和 事件概率 Probabilities of Events(样本空间的子集)对随机现象的数学描述。

 --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和 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.

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.

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)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。

 --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 .

概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 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}。

 --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. 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.

概率分布一般分为两类。 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，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面， 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(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。 正态分布 Normal Distribution是一种常见的连续概率分布。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。

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.

一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为 单变量 Univariate，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。单变量分布给出了单个随机变量取不同替代值的概率; 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。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。

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. 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。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s > 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。

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.]]

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 [math]\displaystyle{ p }[/math] 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

为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个概率质量函数 Probability Mass Function P就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是

[math]\displaystyle{ p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2. }[/math]

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.

相比之下，当一个随机变量从一个连续体中取值时，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。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，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% ，而这一要求对测量仪器的准确性不太敏感。

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.

连续概率分布可以用几种方法来描述。概率密度函数 Probability Density Function描述了任意给定值的无穷小概率，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。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。

文件:Combined Cumulative Distribution Graphs.png

图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve. 左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。

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 [math]\displaystyle{ \infty }[/math] to [math]\displaystyle{ x }[/math] as described by the picture to the right.

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.

累积分布函数指标描述了随机变量不大于给定值的概率; 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。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，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。

[[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。]]

Terminology 术语

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

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.

一些关键的，广泛用于以概率分布为主题的文献中的概念和术语，列出如下。

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.

用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。

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

概率函数 Probability Function：描述离散随机变量的概率分布。

• Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value.

概率质量函数（pmf Probability Mass Function：给出离散随机变量等于某个值的概率的函数。

• Frequency distribution: a table that displays the frequency of various outcomes in a sample.

频率分布 Frequency Distribution：显示样本中各种结果的频率的表格。

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

对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示

• 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.

相对频率分布 Relative Frequency Distribution：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。

• 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.

离散概率分布函数 Discrete Probability Distribution Function：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。

• Cumulative distribution function: function evaluating the probability that [math]\displaystyle{ X }[/math] will take a value less than or equal to [math]\displaystyle{ x }[/math] for a discrete random variable.

累积分布函数 Cumulative distribution function：该函数评估离散随机变量X取小于或等于x的值的概率。

• Categorical distribution: for discrete random variables with a finite set of values.

分类分布 Categorical Distribution：适用于具有有限值集的离散随机变量。

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.

概率密度函数（pdf）Probability Density function：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。

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

由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为

• Continuous probability distribution function: most often reserved for continuous random variables.

连续概率分布函数 Continuous Probability Distribution Function：最常保留的连续随机变量。

• Cumulative distribution function: function evaluating the probability that [math]\displaystyle{ X }[/math] will take a value less than or equal to [math]\displaystyle{ x }[/math] for continuous variable.

累积分布函数 Cumulative distribution function：评估连续变量X取小于或等于x的值的概率的函数。

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.

模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。

• Support: set of values that can be assumed, with non-zero probability, by the random variable.

支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}

• Tail: the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form [math]\displaystyle{ X \gt a }[/math], [math]\displaystyle{ X \lt b }[/math] or a union thereof.

尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X> a，X <b或它们的并集。

• Head: the region where the pmf or pdf is relatively high. Usually has the form [math]\displaystyle{ a \lt X \lt b }[/math].

头部：pmf或pdf较高的区域。通常具有a <X <b的形式

• Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.

期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。

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.

一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。

• 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.

中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。

• Variance: the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.

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.

标准偏差：方差的平方根，因此是色散的另一种度量。

• 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.

对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。

• 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.

偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。

• Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。

Discrete probability distribution 离散概率分布

文件:Discrete probability distrib.svg

图4:The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.

文件:Discrete probability distribution.svg

图5:The cdf of a discrete probability distribution, ...离散概率分布

文件:Normal probability distribution.svg

图6:... of a continuous probability distribution, ...连续概率分布

文件:Mixed probability distribution.svg

图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分

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

离散概率分布是可以具有可数数量的值的概率分布。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。例如，如果[math]\displaystyle{ \operatorname{P}(X=n) = \tfrac{1}{2^n} }[/math] for n = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。

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. 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。

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

[math]\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. }[/math]

This recovers the definition given above.

一个可测量的函数 [math]\displaystyle{ X \colon A \to B }[/math] 在一个概率空间中 [math]\displaystyle{ (A, \mathcal A, P) }[/math] and 和一个可测量空间 [math]\displaystyle{ (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]\displaystyle{ X }[/math]的测量意味着单例集的原像是可测量的 i.e., [math]\displaystyle{ X^{-1}(\{b\}) \in \mathcal A }[/math] 对于所有的[math]\displaystyle{ b \in B }[/math]. 后者需要包括概率质量函数 [math]\displaystyle{ f_X \colon X(A) \to \mathbb R }[/math] via [math]\displaystyle{ f_X(b):=P(X^{-1}(\{b\})) }[/math]. 由于不相交集的原像不相交

[math]\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. }[/math]

这包含了上面所提到的定义

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.

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 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.

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 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

[math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega) }[/math]

except on a set of probability zero, where [math]\displaystyle{ 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. 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。

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

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

一个随机变量[math]\displaystyle{ X }[/math] 有一个连续的概率分布，如果这有一个函数[math]\displaystyle{ f: \mathbb{R} \rightarrow [0, \infty) }[/math] 对于每一个区间[math]\displaystyle{ I \subset \mathbb{R} }[/math] [math]\displaystyle{ I }[/math]的概率 [math]\displaystyle{ X }[/math]是[math]\displaystyle{ f }[/math] 在 [math]\displaystyle{ I }[/math]上的积分. 例如, 如果[math]\displaystyle{ I = [a, b] }[/math] 可得到:

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

In particular, the probability for [math]\displaystyle{ X }[/math] to take any single value [math]\displaystyle{ a }[/math] (that is [math]\displaystyle{ 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]\displaystyle{ F(x) = \operatorname{P}\left[-\infty \lt 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

• [math]\displaystyle{ F(x) }[/math] is non-decreasing;
• [math]\displaystyle{ 0 \le F(x) \le 1 }[/math];
• [math]\displaystyle{ \lim_{x \rightarrow -\infty} F(x) = 0 }[/math] and [math]\displaystyle{ \lim_{x \rightarrow \infty} F(x) = 1 }[/math];
• [math]\displaystyle{ P(a \le X \lt b) = F(b) - F(a) }[/math]; and
• [math]\displaystyle{ F(x) }[/math] is continuous (due to the Riemann integral properties).

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

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 [math]\displaystyle{ X }[/math] is such an absolutely continuous random variable, then it has a probability density function [math]\displaystyle{ f(x) }[/math], and its probability of falling into a Lebesgue-measurable set [math]\displaystyle{ A \subset \mathbb{R} }[/math] is:

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

where [math]\displaystyle{ \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]\displaystyle{ \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, rather than absolutely continuous. These distributions are the ones [math]\displaystyle{ \mu }[/math] such that [math]\displaystyle{ \mu\{x\}\,=\,0 }[/math] for all [math]\displaystyle{ \,x }[/math]. 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.

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。

Kolmogorov definition 柯尔莫哥洛夫的定义

Main articles: Probability space and Probability measure

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

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]\displaystyle{ X_*\mathbb{P} = \mathbb{P}X^{-1} }[/math]的概率量度

Random number generation 随机数生成

Main article: 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 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.

大多数算法基于 伪随机数生成器 Pseudorandom Number Generator，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]\displaystyle{ U }[/math] has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some [math]\displaystyle{ 0 \lt p \lt 1 }[/math], we define

[math]\displaystyle{ {\displaystyle X ={\begin{cases}1,&{\mbox{if }}U\lt p\\0,&{\mbox{if }}U\geq p\end{cases}}} }[/math]

so that

[math]\displaystyle{ \textrm{P}(X=1) = \textrm{P}(U\lt 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{ {\displaystyle X ={\begin{cases}1,&{\mbox{if }}U\lt p\\0,&{\mbox{if }}U\geq p\end{cases}}} }[/math]

因此

[math]\displaystyle{ \textrm{P}(X=1) = \textrm{P}(U\lt 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]\displaystyle{ p }[/math]. 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]\displaystyle{ F }[/math] of a continuous random variable, a continuous random variable must be constructed. [math]\displaystyle{ F^{inv} }[/math], an inverse function of [math]\displaystyle{ F }[/math], relates to the uniform variable [math]\displaystyle{ U }[/math]:

[math]\displaystyle{ {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]\displaystyle{ F^{inv} }[/math]，F的反函数，涉及均匀变量U： [math]\displaystyle{ {U\leq F(x)} = {F^{inv}(U)\leq x}. }[/math]

For example, suppose a random variable that has an exponential distribution [math]\displaystyle{ 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]\displaystyle{ F(x) = 1 - e^{-\lambda x} }[/math] 的随机变量。

[math]\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} }[/math]

所以[math]\displaystyle{ F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u) }[/math] 并且如果 [math]\displaystyle{ U }[/math] 有一个[math]\displaystyle{ U(0,1) }[/math] 分布, 然后随机变量 [math]\displaystyle{ X }[/math] 被定义为 [math]\displaystyle{ X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U) }[/math]. 这里有一个指数分布 [math]\displaystyle{ \lambda }[/math].

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

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.

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 Normal Distribution，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 normally distributed

对数正态分布 Log-normal Distribution，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 exponentially distributed; the prototypical power law distribution

帕累托分布 Pareto Distribution，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 Discrete Uniform Distributed，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 Continuous Uniform Distributed，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 Bernoulli Distribution，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 occurrences

二项式分布 Binomial Distribution ，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 Negative Binomial Distribution，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 Geometric Distribution，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 Hypergeometric Distribution，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 Beta-binomial Distribution，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 Categorical Distribution（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 Multinomial Distribution；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 Multivariate Distribution，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 Poisson Distribution，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 Exponential Distribution，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 Gamma Distribution，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 Rayleigh Distribution，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 Rice Distribution，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 Chi-squared Distribution，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 variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student's t-test)

学生t分布 Student‘s t Distribution，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 variables; useful e.g. for inferences that involve comparing variances or involving R-squared (the squared correlation coefficient)

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

Main article: 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 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

• Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a Poisson distribution or exponential distribution, the 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 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 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

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

• The cache language models and other 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]\displaystyle{ P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 }[/math], probability that the particle's position x will be in the interval a ≤ x ≤ b in dimension one, and a similar triple integral in dimension three. This is a key principle of quantum mechanics.

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]\displaystyle{ 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.

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 distributions such as tropical cyclones, hail, time in between events, etc.

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

• 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

External links

• 模板:Springer
• Field Guide to Continuous Probability Distributions, Gavin E. Crooks.