How Do You Use Intrinsic Functions to Sample from a Normal Distribution?

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Discussion Overview

The discussion revolves around methods for sampling random numbers from a normal distribution, particularly in the context of Monte Carlo simulations. Participants explore various approaches, including the use of intrinsic functions in software and mathematical techniques involving uniform random variables.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant recalls having taken a Monte Carlo class and expresses difficulty in sampling from a normal distribution due to the non-elementary nature of the normal distribution's cumulative distribution function (CDF).
  • Another participant proposes a method using two independent uniform random variables, U1 and U2, to generate standard normal random variables through specific mathematical transformations involving logarithms and trigonometric functions.
  • There is a request for clarification on what is meant by "intrinsic normal distribution function," indicating a need for understanding the term "intrinsic" in this context.
  • A participant reiterates the challenge of using the normal CDF directly due to computational impracticalities and suggests that the law of large numbers can be leveraged, noting that uniform random variables can serve as a foundation for generating other distributions.
  • Another participant clarifies that "intrinsic" refers to built-in functions in software that can compute the normal distribution directly.

Areas of Agreement / Disagreement

Participants express varying levels of familiarity with the topic, and while some methods are proposed, there is no consensus on a single best approach for sampling from a normal distribution. The discussion remains open with multiple viewpoints presented.

Contextual Notes

Some limitations include the dependence on specific software capabilities for intrinsic functions and the unresolved nature of the mathematical steps involved in sampling from the normal distribution.

Watts
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I had a Monte Carlo class many moons ago. I was wondering if some one could jog my memory on how to sample random numbers from a normal distribution. I could do it but the normal distributions CDF is a non elementary integral. I can't get past that part.
 
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Let U1 and U2 be two independent uniforrm random variables over the unit square. Then two independent standard normal random variables can be generated as [itex]N_1 = \sqrt{-2 \log (U_1)} \sin (2\pi U_2)[/itex] and [itex]N_2 = \sqrt{-2 \log (U_1)} \cos (2\pi U_2)[/itex].

P.S. Any software with an intrinsic normal distribution function will also do the trick.
 
Last edited:
Explain

Explain intrinsic normal distribution function(the intrinsic part).
 
Last edited:
Watts said:
I had a Monte Carlo class many moons ago. I was wondering if some one could jog my memory on how to sample random numbers from a normal distribution. I could do it but the normal distributions CDF is a non elementary integral. I can't get past that part.
The obvious method would be to generate uniform ramdon numbers on [0,1] then invert the normal CDF, but that is not computationally practical. What is often done is using the law of large numbers. The average of a large number of nonpathological random variables will be normal. Uniform randoms on [0,1] work well and are often the basis for other distributions. Also as was mentioned one could use a program/library that includes a random normal generator.
 
Watts said:
Explain intrinsic normal distribution function(the intrinsic part).

"Intrinsic" here just means that the software has a built in function that will calculate the normal distribution for you.
 

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