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In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].<ref name=":02">{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}</ref><ref>{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}</ref> It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).<ref name=":1">{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}</ref>

In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].<ref name=":02">{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}</ref><ref>{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}</ref> It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).<ref name=":1">{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}</ref>

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).

在概率论和统计学中，概率分布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。它是根据样本空间和事件概率(样本空间的子集)对随机现象的数学描述。

在'''<font color="#ff8000">概率论 Probability Theory</font>'''和'''<font color="#ff8000">统计学 Statistics</font>'''中，'''<font color="#ff8000"> 概率分布 Probability Distribution</font>'''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'''<font color="#ff8000"> 样本空间 Sample Space</font>'''和'''<font color="#ff8000"> 事件概率 Probabilities of Events</font>'''(样本空间的子集)对随机现象的数学描述。

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。

概率分布一般分为两类。'''<font color="#ff8000"> 离散概率分布 Discrete Probability Distribution</font>'''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，'''<font color="#ff8000"> 连续概率分布 Continuous Probability Distribution</font>'''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。'''<font color="#ff8000"> 正态分布 Normal Distribution</font>'''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，而样本空间为二维或更多向量空间的分布被称为多变量。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。

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'''<font color="#ff8000"> 单变量 Univariate</font>'''，而样本空间为二维或更多向量空间的分布被称为'''<font color="#ff8000"> 多变量 Multivariate</font>'''。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。

=== Functions for discrete variables 离散变量的函数===

=== '''<font color="#ff8000"> Functions for discrete variables 离散变量函数</font>'''===

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

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

*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.

*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.

*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.

*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.

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

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

*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.

*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.

*'''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.

*'''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.

*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for a discrete random variable.

*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for a discrete random variable.

*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.

*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.

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

=== '''<font color="#ff8000"> Functions for continuous variables 连续变量函数</font>'''===

* '''[[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.

* '''[[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.

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

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

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

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

* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for continuous variable.

* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for continuous variable.

==Discrete probability distribution 离散概率分布==

=='''<font color="#ff8000"> Discrete probability distribution 离散概率分布</font>'''==

这包含了上面所提到的定义

这包含了上面所提到的定义

===Cumulative distribution function===

==='''<font color="#ff8000"> 累积分布函数 Cumulative distribution function</font>'''===

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.

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。

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==

== '''<font color="#ff8000"> [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义</font>'''==

{{Main|Probability space|Probability measure}}

{{Main|Probability space|Probability measure}}

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>

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。

大多数算法基于'''<font color="#ff8000"> 伪随机数生成器 Pseudorandom Number Generator</font>'''，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

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

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。

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===

=== '''<font color="#ff8000">Linear growth (e.g. errors, offsets) 线性增长 </font>'''===

* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution

* [[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===

=== '''<font color="#ff8000"> Exponential growth (e.g. prices, incomes, populations) 指数增长</font>'''===

* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed

* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed

* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution

* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution

=== Uniformly distributed quantities 数量均匀分布===

=== '''<font color="#ff8000"> Uniformly distributed quantities 数量均匀分布</font>'''===

* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)

* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)

* [[Continuous uniform distribution]], for continuously distributed values

* [[Continuous uniform distribution]], for continuously distributed values

=== 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）===

=== 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:

* Basic distributions 基本分布:

** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)

** [[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）

'''<font color="#ff8000"> 伯努利分布 Bernoulli Distribution</font>'''，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

** [[Binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences

** [[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

** [[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

** [[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]]

** [[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]]

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

* 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]]

** [[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]]

** [[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]])

** [[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） ）

'''<font color="#ff8000"> 贝塔二项式分布 Beta-binomial Distribution</font>'''，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 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）===

* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time

* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time

* [[Exponential distribution]], for the time before the next Poisson-type event occurs

* [[Exponential distribution]], for the time before the next Poisson-type event occurs

* [[Gamma distribution]], for the time before the next k Poisson-type events occur

* [[Gamma distribution]], for the time before the next k Poisson-type events occur

=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===

=== 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.

* [[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.

* [[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）===

=== 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]])

* [[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]])

* [[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]])

* [[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]])

* [[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-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

'''<font color="#ff8000">F-分布 F-Distribution</font>'''，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===

=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===

* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]

* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]

* [[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.

* [[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.

* [[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 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

'''<font color="#ff8000">Dirichlet分布 Dirichlet Distribution</font>'''，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 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>

=== Some specialized applications of probability distributions 概率分布的一些特殊应用===

=== Some specialized applications of probability distributions 概率分布的一些特殊应用===