# Talk:Normal distribution

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## Untitled

Would it be fair to say that few if any math majors turn to Wikipedia for help in their chosen field? If so, who exactly is this article written for? Unless they have post-secondary studies in Math, few people would have the knowledge or time to comprehend any of the terms used & these beginners, I would submit, are the vast majority of those who click on this article. We would just like, in layman's terms, an explanation of Normal Distributuion. Instead we've found a long, specialized article written for no one. — Preceding unsigned comment added by 96.55.2.6 (talk) 22:40, 26 March 2013 (UTC)

Agreed. Livingston 08:51, 21 April 2013 (UTC)
Thirded. The article should begin with an intuitive explanation. It is far too technical right from the start. Plantsurfer (talk) 11:12, 21 April 2013 (UTC)
While I am here, an animated figure of the kind shown at right has great potential to communicate what a normal distribution is, but it is a great pity that the values that are contributing to the curve are at discrete, symmetrical intervals, and that they perfectly fit the normal curve right from the outset. That is not how it works. It would be a lot better to have a similar graphic based on a real or realistically modeled data set. Plantsurfer (talk) 11:21, 21 April 2013 (UTC)

You have links to the terms you don't understand. Also, N is an advanced subject in itself. i.e. it can not be simplified without being hollow and meaningless. Read about other types of distributions first if you want simpler examples of that type of math. The reason for the complexity, or rather lack of a comprehensive explanation for it, is that the distribution is not human constructed but an observed reality of life. It just happens to work for many common situations. --5.54.91.60 (talk) 20:03, 21 June 2013 (UTC)

I'm confused. Shouldn't the ERF function be defined as ERF(a,b) = integral between a and b, instead of ERF(x) = integral between -x and +x? This would then allow for the proper definition of the CDF function as ERF(-infinity, x) instead of defining it as a single value function. Maybe the error introduced by using -x instead of -infinity is small.

130.76.64.109 (talk) 16:15, 4 July 2013 (UTC)

## Clarification needed

It should be noted that the Normal Distribution Function comes from the Stirling's approximation applied to the Binomial distribution (deMoivres-Laplace: http://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem.) In the binomial distribution, the probability of "each outcome" is known. That is Binomial distribution builds on that fact that "I can get k successes in n trials where each event has a probability p", and I plot the value of E(k) versus k. When I carry this to the Stirling approximation to form the Normal Distribution function, I assume each independent event has the same "p". What is this "p" that I refer to now in the context of a normal distribution function? In other words are the trials still "Bernoulli"? If yes what is the p used in the context of NDF.

If however one is simply assuming this is "distribution" function and the central limit theorem is just a coincidence, then note that most derivations of the central limit theorem also build from Binomial distribution. Can someone please clarify what is "Bernoulli" about the trails in that case? Is each E(x) associated with x still representing a success of a "Bernoulli outcome" at all??? The literature on this page, and the "central limit theorem" is not clear and is recursive..and always points back to Demoive-Laplace Theorem only.

An independent proof of the "Central limit Theorem", not relying on Binomial distribution would also help clarify this circular reference.

-Alok 11:31, 19 July 2013 (UTC) — Preceding unsigned comment added by Alokdube (talkcontribs)

It should also be noted that wikipedia does not in anyway state that Normal Distribution function is sacrosanct but most text books and academicians tend to do so. However it would be really great if someone can show the assumptions made in the approach. -Alok 23:10, 23 July 2013 (UTC) — Preceding unsigned comment added by Alokdube (talkcontribs)

## The Walsh Hadamard transform and the Normal Distribution

I eventually found this paper on using the Walsh Hadamard transform: [1] Wallace, C. S. 1996. "Fast Pseudorandom Generators for Normal and Exponential Variates." ACM Transactions on Mathematical Software.

I independently discovered the idea myself around 2001. I further showed that by combining the Walsh Hadamard transform with random permutations you can convert arbitrary numerical data into the Gaussian distribution. I am not sure if anyone has any prior claim to that. I have used it to create associative memory algorithms and as a population based method for generating random numbers for Evolutionary Strategies (ES) based algorithms. I am sure it would have other uses. A useful reference is [1] I am pretty sure NVidia got the idea from me (because I sent them an e-mail about it). They did however find the reference to Wallace which I could not find. Maybe you can still find some of my code on the forum of www.freebasic.net but a lot of it is gone from the Internet because no gain. Sean O'Connor

## A simpler formula for the pdf - should it be in this article?

The PDF can be re-arranged to the following form:

$f(x) = \frac{1}{\sigma\sqrt{2\pi e^{(\frac{x-\mu}{\sigma})^2}}} = \frac{1}{\sigma\sqrt{2\pi e^{Z^2}}}$

where Z is the Standard score (number of standard deviations from the mean). This makes it pretty obvious that the pdf is maximal when $\sigma$ is small (narrow distribution) and when $Z$ is small (towards the center of the distribution). I find this notation way simpler and more intuitive than the standard formula for the pdf. Should we include it in the main article (and where?) for the pedagogical purpose? — Preceding unsigned comment added by 129.215.5.255 (talk) 10:46, 30 October 2013 (UTC)

## Normal Sum Theorem

The normal sum theorem for the sum of two normal variates is discussed in Lemons, Don (2002). An Introduction to Stochastic Processes in Physics. John Hopkins. p. 34. ISBN 0-8018-6867-X.. The proof of the theorem shows that the variance for the sum is the sum of the two variances. However, this doesn't prove the distribution for the sum is a normal distribution since more than one distribution can have the same variance. --Jbergquist (talk) 06:18, 30 November 2013 (UTC)

If x and y have normal distributions with zero means and standard deviations of σ and s respectively, the probability density for all combinations of x and y is just the product of the two normal distributions. One can then show that the probability distribution for z=x+y is a normal distribution with mean zero and variance σ2+s2. The proof involves transforming the joint probability density to a new set of variables, z=x+y and w=x-y, then integrating over all values of w to get the probability density for z. --Jbergquist (talk) 02:50, 2 December 2013 (UTC)

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