Deduction of Gauss' Normal Distribution Function

In summary, David is a new member seeking help in finding the formal derivation of Gauss' Normal Distribution Function. He has read many statistics books but has not found it there. He clarifies that it is out of curiosity, not for homework purposes. Another member suggests that David may mean derivation instead of deduction, and provides a link to the Central Limit Theorem. However, this does not derive the general equation for the normal distribution. Another member provides a link to a PDF that derives an equation for a specific problem. It is also mentioned that the normal distribution can be derived as a limit of the binomial distribution, and that the historical motivation for the normal distribution comes from the limit of binomials and the Central Limit Theorem.
  • #1
davidsousarj
3
0
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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  • #2
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.
 
  • #3
Hi Stephen, thanks for the ansωer. I didn't knoω this theorem. Very interesting.
 
  • #4
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.
 
  • #5
thanks, guicortei.
 
  • #6
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.
 
  • #7
You can also derive the normal distribution as the limit of the binomial one.
 
  • #8
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 )
 

What is Gauss' Normal Distribution Function?

Gauss' Normal Distribution Function, also known as the Gaussian Distribution or the Bell Curve, is a probability distribution that is commonly used in statistics to model continuous random variables. It is characterized by its symmetrical, bell-shaped curve and is used to describe many natural phenomena.

Who is responsible for the deduction of Gauss' Normal Distribution Function?

The deduction of Gauss' Normal Distribution Function is credited to German mathematician and physicist Carl Friedrich Gauss, who first described it in his work "Theoria Motus Corporum Coelestium" in 1809.

What are the key properties of Gauss' Normal Distribution Function?

The key properties of Gauss' Normal Distribution Function include its symmetrical, bell-shaped curve, its mean and standard deviation, and its total area under the curve being equal to 1. It also follows the 68-95-99.7 rule, where approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

How is Gauss' Normal Distribution Function used in real-world applications?

Gauss' Normal Distribution Function is widely used in various fields such as finance, psychology, and biology, to name a few. It is used to model a wide range of natural phenomena, including heights and weights of individuals, test scores, and stock prices. It is also used in quality control and process control to determine if a process is within acceptable limits.

What are the limitations of Gauss' Normal Distribution Function?

While Gauss' Normal Distribution Function is widely used and highly versatile, it has some limitations. It assumes that the data is normally distributed and can be skewed if the data is not symmetric. It also cannot be used for discrete data, as it is a continuous distribution. In some cases, alternative distributions may be a better fit for the data.

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