Editing Pi (section)

=== Gaussian integrals ===
[[File:E^(-x^2).svg|thumb|A graph of the [[Gaussian function]] {{math|1=''ƒ''(''x'') = ''e''{{sup|−''x''{{sup|2}}}}}}. The coloured region between the function and the ''x''-axis has area {{math|{{sqrt|π}}}}.|left]]

The fields of [[probability]] and [[statistics]] frequently use the [[normal distribution]] as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.<ref>Feller, W. ''An Introduction to Probability Theory and Its Applications, Vol. 1'', Wiley, 1968, pp. 174–190.</ref> The [[Gaussian function]], which is the [[probability density function]] of the normal distribution with [[mean]] {{math|μ}} and [[standard deviation]] {{math|σ}}, naturally contains {{pi}}:<ref name="GaussProb">{{harvnb|Bronshteĭn|Semendiaev|1971|pp=106–107, 744, 748}}.</ref>

<math display=block>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}.</math>

The factor of <math>\tfrac{1}{\sqrt{2\pi}}</math> makes the area under the graph of {{math|''f''}}  equal to one, as is required for a probability distribution. This follows from a [[integration by substitution|change of variables]] in the [[Gaussian integral]]:{{r|GaussProb}}

<math display=block>\int_{-\infty}^\infty e^{-u^2} \, du=\sqrt{\pi}</math>

which says that the area under the basic [[bell curve]] in the figure is equal to the square root of {{pi}}.

The [[central limit theorem]] explains the central role of normal distributions, and thus of {{pi}}, in probability and statistics. This theorem is ultimately connected with the [[#Fourier transform and Heisenberg uncertainty principle|spectral characterization]] of {{pi}} as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function.{{sfn|Dym|McKean|1972|loc=Section 2.7}} Equivalently, {{pi}} is the unique constant making the Gaussian normal distribution {{math|''e''{{sup|−π''x''{{sup|2}}}}}} equal to its own Fourier transform.<ref>{{cite book |first1=Elias |last1=Stein |first2=Guido |last2=Weiss |title=Fourier analysis on Euclidean spaces |year=1971 |publisher=Princeton University Press |page=6 |author1-link=Elias Stein}}; Theorem 1.13.</ref> Indeed,  according to {{harvtxt|Howe|1980}}, the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.{{r|howe}}