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There is no consensus on how ''Monte Carlo'' should be defined. For example, Ripley<ref name=Ripley>{{harvnb|Ripley|1987}}</ref> defines most probabilistic modeling as ''[[stochastic simulation]]'', with ''Monte Carlo'' being reserved for [[Monte Carlo integration]] and Monte Carlo statistical tests. [[Shlomo Sawilowsky|Sawilowsky]]<ref name=Sawilowsky>{{harvnb|Sawilowsky|2003}}</ref> distinguishes between a [[simulation]], a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Examples:

There is no consensus on how ''Monte Carlo'' should be defined. For example, Ripley<ref name=Ripley>{{harvnb|Ripley|1987}}</ref> defines most probabilistic modeling as ''[[stochastic simulation]]'', with ''Monte Carlo'' being reserved for [[Monte Carlo integration]] and Monte Carlo statistical tests. [[Shlomo Sawilowsky|Sawilowsky]]<ref name=Sawilowsky>{{harvnb|Sawilowsky|2003}}</ref> distinguishes between a [[simulation]], a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Examples:

*Simulation: Drawing ''one'' pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.

*Simulation: Drawing ''one'' pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.

*Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

*Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

*Monte Carlo simulation: Drawing <nowiki>''</nowiki>a large number<nowiki>''</nowiki> of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a <nowiki>''</nowiki>Monte Carlo simulation<nowiki>''</nowiki> of the behavior of repeatedly tossing a coin.

*Monte Carlo simulation: Drawing <nowiki>''</nowiki>a large number<nowiki>''</nowiki> of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a <nowiki>''</nowiki>Monte Carlo simulation<nowiki>''</nowiki> of the behavior of repeatedly tossing a coin.

 

*蒙特卡罗模拟法:绘制伪随机均匀的“大量”变量的区间[0,1],或曾在许多不同的时间,和赋值小于或等于0.50作为正面和大于0.50反面,是一个“蒙特卡罗模拟”的行为多次掷硬币。

对于如何定义蒙特卡洛还没有达成共识。例如，Ripley将大多数概率建模定义为随机模拟，蒙特卡罗保留用于蒙特卡罗积分和蒙特卡罗统计检验。Sawilowsky[54]区分了模拟、蒙特卡罗方法和蒙特卡罗模拟:蒙特卡罗方法是一种可以用来解决数学或统计问题的技术，蒙特卡罗模拟使用重复抽样来获得某些现象(或行为)的统计特性。例如：

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]].  RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10⁷ random numbers.<ref>{{cite journal|last1=Route|first1=Matthew|title=Radio-flaring Ultracool Dwarf Population Synthesis|journal=The Astrophysical Journal|date=August 10, 2017|volume=845|issue=1|page=66|doi=10.3847/1538-4357/aa7ede|arxiv=1707.02212|bibcode=2017ApJ...845...66R|s2cid=118895524}}</ref>

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]].  RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10⁷ random numbers.<ref>{{cite journal|last1=Route|first1=Matthew|title=Radio-flaring Ultracool Dwarf Population Synthesis|journal=The Astrophysical Journal|date=August 10, 2017|volume=845|issue=1|page=66|doi=10.3847/1538-4357/aa7ede|arxiv=1707.02212|bibcode=2017ApJ...845...66R|s2cid=118895524}}</ref>

=== Monte Carlo simulation versus "what if" scenarios ===

=== Monte Carlo simulation versus "what if" scenarios ===