Editing Normal distribution (section)

=== Standard normal distribution ===
The simplest case of a normal distribution is known as the '''standard normal distribution''' or '''unit normal distribution'''. This is a special case when <math display=inline>\mu=0</math> and <math display=inline>\sigma^2 =1</math>, and it is described by this [[probability density function]] (or density):<ref>{{harvtxt|Mood|1950|loc=[https://archive.org/details/introductiontoth0000alex/page/112/mode/2up?q=%22standard+normal+distribution%22 p. 112]}} explicitly defines the ''standard normal distribution''. In contrast, {{harvtxt|Hoel|1947}} explicitly defines the ''standard normal curve'' [https://archive.org/details/in.ernet.dli.2015.263186/page/n41/mode/2up?q=%22standard+normal+curve%22 (p. 33)] and introduces the term ''standard normal distribution'' [https://archive.org/details/in.ernet.dli.2015.263186/page/n77/mode/2up?q=%22standard+normal+distribution%22 (p. 69)].</ref>
<math display="block">\varphi(z) = \frac{e^{-z^2/2}}{\sqrt{2\pi}}\,.</math>
The variable {{tmath|z}} has a mean of 0 and a variance and standard deviation of 1. The density <math display=inline>\varphi(z)</math> has its peak <math display="inline">\frac{1}{\sqrt{2\pi}}</math> at <math display=inline>z=0</math> and [[inflection point]]s at <math display=inline>z=+1</math> and {{tmath|1=z=-1}}.

Although the density above is most commonly known as the ''standard normal,'' a few authors have used that term to describe other versions of the normal distribution. [[Carl Friedrich Gauss]], for example, once defined the standard normal as
<math display=block>\varphi(z) = \frac{e^{-z^2}}{\sqrt\pi},</math>
which has a variance of {{tmath|\frac{{mset|1}}{{mset|2}}}}, and [[Stephen Stigler]]<ref>{{harvtxt |Stigler |1982 }}</ref> once defined the standard normal as
<math display=block>\varphi(z) = e^{-\pi z^2},</math>
which has a simple functional form and a variance of <math display=inline>\sigma^2 = \frac {1}{2\pi}.</math>