

The Normal Distribution: A derivation from basic principles [pdf] - nkron
http://courses.ncssm.edu/math/Talks/PDFS/normal.pdf

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csense
Thanks for posting this. I've been looking for intuitions about what's so
special about the normal distribution.

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widdma
While this is a nice little derivation, the reason the normal distribution is
important (and common) is the Central limit theorem[1].

This loosely means that if you have n samples that are independently drawn
from the same distribution, their sum approaches a normal distribution for
large n. The generality comes from the fact that this happens (almost)
independently of distribution the samples come from.

[1][https://en.wikipedia.org/wiki/Central_limit_theorem](https://en.wikipedia.org/wiki/Central_limit_theorem)

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stiff
This is often given as the reason in the textbooks, but it actually is a
sloppy argument, for there is no reason to presuppose sums of independent
variables (they do not have to be drawn from the same distribution in some
variants of the CLT) are so ubiquitous in nature. A more convincing reason is
given by:

[http://en.wikipedia.org/wiki/Maximum_entropy_probability_dis...](http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution)

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mturmon
Well, I think textbooks often give the maxent justification as well, but
probably neither is strictly superior to the other.

Often, where a measurement is perturbed by a litany of physically unlinked
disturbances (e.g., antenna noise), the CLT argument is compelling. In a case
like antenna or front-end electronics noise, justifying the maxent argument
would require buying in to an almost mystical belief by comparison.

Interestingly, the two seem to have a deep connection related to
thermodynamics, e.g.
[http://en.wikipedia.org/wiki/Thermal_fluctuations](http://en.wikipedia.org/wiki/Thermal_fluctuations),
where you can see both as expansions of a configuration count up to second
order in the exponent.

This is manifest in the detectable remnants of the Big Bang, which are a
particular realization of a Gaussian random field that was present due to
quantum fluctuations of a primordial system. It's funny to think of the
universe as a draw from a random number generator.

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quarterwave
Isn't the key point here about isotropy, similar to the velocities in the
ideal gas?

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mturmon
I think you're right, but it's really isotropy in combination with
independence of x and y. Isotropy means you really are dealing with a function
of r where r^2 = x^2 + y^2, and independence means the resulting function
p(x,y) must factor as p(x) * p(y). This then forces the exponential form.

