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For example, suppose a random variable that has an exponential distribution <math>F(x) = 1 - e^{-\lambda x}</math> must be constructed.

For example, suppose a random variable that has an exponential distribution <math>F(x) = 1 - e^{-\lambda x}</math> must be constructed.

例如，假设必须构造一个具有指数分布<math>F(x) = 1 - e^{-\lambda x}</math> 的随机变量。

例如，假设必须构造一个具有指数分布<math>F(x) = 1 - e^{-\lambda x}</math> 的随机变量。

 

<math>\begin{align}

<math>\begin{align}

F(x) = u &\Leftrightarrow 1-e^{-\lambda x} = u \\ &\Leftrightarrow e^{-\lambda x } = 1-u \\&\Leftrightarrow -\lambda x = \ln(1-u) \\ &\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)

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)

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

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

 

A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.

A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.

统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。

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==

== Common probability distributions and their applications 共同概率分布及其应用==

{{Main list|List of probability distributions}}

{{Main list|List of probability distributions}}

 

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.

概率分布和内蕴其中的随机变量的概念是概率论和统计学领域的基础。几乎任何可以在群体中测量的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）;几乎所有测量都存在一定的固有误差；在物理学中，从气体动力学理论到基本粒子的量子力学描述都用概率论来描述。出于各种原因，简单数字通常很难表述一个量量，而用概率分布表述通常更合适。

概率分布和内蕴其中的随机变量的概念是概率论和统计学领域的基础。几乎任何可以在群体中测量的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）;几乎所有测量都存在一定的固有误差；在物理学中，从气体动力学理论到基本粒子的量子力学描述都用概率论来描述。出于各种原因，简单数字通常很难表述一个量量，而用概率分布表述通常更合适。

 

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.)

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。

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。

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

=== '''<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

'''<font color="#ff8000"> 正态分布（高斯分布 Normal Distribution</font>'''，适用于单个变量；是最常用的连续分布

'''<font color="#ff8000"> 正态分布（高斯分布 Normal Distribution</font>'''，适用于单个变量；是最常用的连续分布

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

=== '''<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

'''<font color="#ff8000">对数正态分布 Log-normal Distribution</font>'''，适用于对数为正态分布的单个变量

'''<font color="#ff8000">对数正态分布 Log-normal Distribution</font>'''，适用于对数为正态分布的单个变量

=== 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===

=== 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===

 

* 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)