Deduction of Gauss' Normal Distribution Function

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Discussion Overview

The discussion revolves around the formal deduction of Gauss' Normal Distribution Function, exploring its derivation, historical context, and related theorems such as the Central Limit Theorem. Participants express curiosity about the mathematical foundations and implications of the normal distribution in statistics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • David seeks a formal deduction of the Gaussian Normal Distribution Function, expressing curiosity rather than a homework-related inquiry.
  • One participant suggests that David may be looking for a "derivation" rather than a "deduction," linking it to the Central Limit Theorem as a relevant theorem.
  • Another participant provides a link to a PDF claiming to contain the derivation, although it is noted that it addresses a specific problem rather than the general equation.
  • There is a discussion about the Central Limit Theorem, which states that the mean of a large number of independent random variables is approximately normally distributed, but this does not derive the normal distribution equation itself.
  • One participant mentions that the normal distribution can also be derived as the limit of the binomial distribution.
  • The historical context of the Gaussian distribution is mentioned, indicating its connection to the limit of binomial distributions and the Central Limit Theorem.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the deduction or derivation of the normal distribution, with some emphasizing the historical context and others focusing on specific mathematical derivations. The discussion remains unresolved regarding the formal deduction of the Gaussian Normal Distribution Function.

Contextual Notes

Participants highlight limitations in understanding the derivation and its application to experimental measures, indicating that the relationship between theoretical distributions and empirical data is not fully demonstrated.

davidsousarj
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Hello, I'm David. I'm a new member here.

Could anyone of you help me? Where can i find the formal deduction of Gauss' Normal Distribution Function? I've read a lot of statistics books and never found that. Where that comes from?

It's just curiosity, not homework.

P.S.: sorry about my english, I'm brazilian and my english is a little poor.
 
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Instead of "deduction", you may mean "derrivation". A derrivation would be a theorem of the form "If ... then the distribution has the following formula..." and the formula would be the one that is used for the gaussian distribution. The most famous theorem of this type is the Central Limit Theorem http://en.wikipedia.org/wiki/Central_limit_theorem .

If you are asking why the formula for the density for a gaussian distribution has the name "gaussian" or "normal", that's a historical question. There isn't any mathematical derrivation for it.
 
Hi Stephen, thanks for the ansωer. I didn't knoω this theorem. Very interesting.
 
Hi David.. I know this topic is very OLD (about 1.5 year), but when searching on google for exactly what you want, i was delivered to this forum... with no answer... so I kept searching and I found the derivation... you can find it here:

http://www.planetmathematics.com/DerNorm.pdf

P.S.: sorry about my english, I'm brazilian too.
 
thanks, guicortei.
 
The central limit theorem says that the arithmetic mean of a sufficiently large number of iterates of independent random variables will be approximately normally distributed,
but that doesn't derive the normal distribution equation in general.

And the pdf, just derives (again) an equation for a particular problem.

http://www.ams.org/journals/tran/1922-024-02/S0002-9947-1922-1501218-2/S0002-9947-1922-1501218-2.pdf

But I can't demostrate is that in a experiment the measures follow that distribution.
 
You can also derive the normal distribution as the limit of the binomial one.
 
The history of the Gaussian distribution is that it was motivated by the limit of binomials and the Central Limit Theorem (see http://www.sjsu.edu/faculty/watkins/randovar.htm )
 

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