Editing Normal distribution (section)

=== Development ===
Some authors<ref>{{harvtxt |Johnson |Kotz |Balakrishnan |1994 |p=85 }}</ref><ref>{{harvtxt |Le Cam | Lo Yang |2000 |p=74 }}</ref> attribute the discovery of the normal distribution to [[de Moivre]], who in 1738{{NoteTag|De Moivre first published his findings in 1733, in a pamphlet ''Approximatio ad Summam Terminorum Binomii {{math|(''a'' + ''b'')<sup>''n''</sup>}} in Seriem Expansi'' that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example {{harvtxt |Walker |1985 }}.}} published in the second edition of his ''[[The Doctrine of Chances]]'' the study of the coefficients in the [[binomial expansion]] of {{math|(''a'' + ''b'')<sup>''n''</sup>}}. De Moivre proved that the middle term in this expansion has the approximate magnitude of <math display=inline>2^n/\sqrt{2\pi n}</math>, and that "If ''m'' or {{sfrac|1|2}}''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is <math display=inline>-\frac{2\ell\ell}{n}</math>."<ref>De Moivre, Abraham (1733), Corollary I – see {{harvtxt |Walker |1985 |p=77 }}</ref> Although this theorem can be interpreted as the first obscure expression for the normal probability law, [[Stephen Stigler|Stigler]] points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.<ref>{{harvtxt |Stigler |1986 |loc=[https://archive.org/details/historyofstatist00stig/page/76/mode/2up?q=%22de+moivre%22 p. 76] }}</ref>

[[File:Carl Friedrich Gauss.jpg|thumb|180px|left|[[Carl Friedrich Gauss]] discovered the normal distribution in 1809 as a way to rationalize the [[method of least squares]].]]

In 1823 [[Gauss]] published his monograph <span title="Theory of the Combination of Observations Least Subject to Errors">"''Theoria combinationis observationum erroribus minimis obnoxiae''"</span> where among other things he introduces several important statistical concepts, such as the [[method of least squares]], the [[method of maximum likelihood]], and the ''normal distribution''. Gauss used ''M'', {{nobr|''M''′}}, {{nobr|''M''′′, ...}} to denote the measurements of some unknown quantity&nbsp;''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability {{math|''φ''(''M'' − ''V'') · ''φ''(''M′'' − ''V'') · ''φ''(''M''′′ − ''V'') · ...}} of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.{{NoteTag|"It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." — {{harvtxt |Gauss |1809 |loc=section 177 }} }} Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:<ref>{{harvtxt |Gauss |1809 |loc=section 177 }}</ref>
<math display=block>
    \varphi\mathit{\Delta} = \frac h {\surd\pi} \, e^{-\mathrm{hh}\Delta\Delta},
</math> <!-- please do not modify this formula; its spacing and style follow the original as closely as possible -->
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the [[non-linear least squares|non-linear]] [[weighted least squares]] method.<ref>{{harvtxt |Gauss |1809 |loc=section 179 }}</ref>

[[File:Pierre-Simon Laplace.jpg|thumb|180px|right| [[Pierre-Simon Laplace]] proved the [[central limit theorem]] in 1810, consolidating the importance of the normal distribution in statistics.]]

Although Gauss was the first to suggest the normal distribution law, [[Laplace]] made significant contributions.{{NoteTag|"My custom of terming the curve the Gauss–Laplacian or ''normal'' curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from {{harvtxt |Pearson |1905 |p=189 }} }} It was Laplace who first posed the problem of aggregating several observations in 1774,<ref>{{harvtxt |Laplace |1774 |loc = Problem III }}</ref> although his own solution led to the [[Laplacian distribution]]. It was Laplace who first calculated the value of the [[Gaussian integral|integral {{math|∫ ''e''<sup>−''t''<sup>2</sup></sup>&nbsp;''dt'' {{=}} {{sqrt|{{pi}}}}}}]] in 1782, providing the normalization constant for the normal distribution.<ref>{{harvtxt |Pearson |1905 |p=189 }}</ref> For this accomplishment, Gauss acknowledged the priority of Laplace.<ref>{{harvtxt |Gauss |1809 |loc=section 177 }}</ref> Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental [[central limit theorem]], which emphasized the theoretical importance of the normal distribution.<ref>{{harvtxt |Stigler |1986 |p=144 }}</ref>

It is of interest to note that in 1809 an Irish-American mathematician [[Robert Adrain]] published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss.<ref>{{harvtxt |Stigler |1978 |p=243 }}</ref> His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by [[Cleveland Abbe|Abbe]].<ref>{{harvtxt |Stigler |1978 |p=244 }}</ref>

In the middle of the 19th century [[James Clerk Maxwell|Maxwell]] demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:{{sfnp|Maxwell|1860|p=23}} The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x''&nbsp;+&nbsp;''dx'' is
<math display=block>
    \operatorname{N} \frac{1}{\alpha\;\sqrt\pi}\; e^{-\frac{x^2}{\alpha^2}} \, dx
</math> <!-- please do not modify this formula; its spacing and style follow the original as close as possible -->