Editing Normal distribution (section)

=== Symmetries and derivatives ===
The normal distribution with density <math display=inline>f(x)</math> (mean {{tmath|\mu}} and variance <math display=inline>\sigma^2 > 0</math>) has the following properties:
* It is symmetric around the point <math display=inline>x=\mu,</math> which is at the same time the [[mode (statistics)|mode]], the [[median]] and the [[mean]] of the distribution.<ref name="Patel">{{harvtxt |Patel |Read |1996 |loc=[2.1.4] }}</ref>
* It is [[unimodal]]: its first [[derivative]] is positive for <math display=inline>x<\mu,</math> negative for <math display=inline>x>\mu,</math> and zero only at <math display=inline>x=\mu.</math>
* The area bounded by the curve and the {{tmath|x}}-axis is unity (i.e. equal to one).
* Its first derivative is <math display=inline>f'(x)=-\frac{x-\mu}{\sigma^2} f(x).</math>
* Its second derivative is <math display=inline>f''(x) =  \frac{(x-\mu)^2 - \sigma^2}{\sigma^4} f(x).</math>
* Its density has two [[inflection point]]s (where the second derivative of {{tmath|f}} is zero and changes sign), located one standard deviation away from the mean, namely at <math display=inline>x=\mu-\sigma</math> and <math display=inline>x=\mu+\sigma.</math><ref name="Patel" />
* Its density is [[logarithmically concave function|log-concave]].<ref name="Patel" />
* Its density is infinitely [[differentiable]], indeed [[supersmooth]] of order 2.<ref>{{harvtxt |Fan |1991 |p=1258 }}</ref>

Furthermore, the density {{tmath|\varphi}} of the standard normal distribution (i.e. <math display=inline>\mu=0</math> and <math display=inline>\sigma=1</math>) also has the following properties:
* Its first derivative is <math display=inline>\varphi'(x)=-x\varphi(x).</math>
* Its second derivative is <math display=inline>\varphi''(x)=(x^2-1)\varphi(x)</math>
* More generally, its {{mvar|n}}th derivative is <math display=inline>\varphi^{(n)}(x) = (-1)^n\operatorname{He}_n(x)\varphi(x),</math> where <math display=inline>\operatorname{He}_n(x)</math> is the {{mvar|n}}th (probabilist) [[Hermite polynomial]].<ref>{{harvtxt |Patel |Read |1996 |loc=[2.1.8] }}</ref>
* The probability that a normally distributed variable {{tmath|X}} with known {{tmath|\mu}} and <math display=inline>\sigma^2</math> is in a particular set, can be calculated by using the fact that the fraction <math display=inline>Z = (X-\mu)/\sigma</math> has a standard normal distribution.