Editing Normal distribution (section)

=== Moments ===
{{See also|List of integrals of Gaussian functions}}
The plain and absolute [[moment (mathematics)|moments]] of a variable {{tmath|X}} are the expected values of <math display=inline>X^p</math> and <math display=inline>|X|^p</math>, respectively. If the expected value {{tmath|\mu}} of {{tmath|X}} is zero, these parameters are called ''central moments;'' otherwise, these parameters are called ''non-central moments.'' Usually we are interested only in moments with integer order {{tmath|p}}.

If {{tmath|X}} has a normal distribution, the non-central moments exist and are finite for any {{tmath|p}} whose real part is greater than&nbsp;−1. For any non-negative integer {{tmath|p}}, the plain central moments are:<ref>{{cite book|last1=Papoulis|first1=Athanasios|title=Probability, Random Variables and Stochastic Processes|page=148|edition=4th}}</ref>
<math display=block>
    \operatorname{E}\left[(X-\mu)^p\right] =
      \begin{cases}
        0 & \text{if }p\text{ is odd,} \\
        \sigma^p (p-1)!! & \text{if }p\text{ is even.}
      \end{cases}
</math>
Here <math display=inline>n!!</math> denotes the [[double factorial]], that is, the product of all numbers from {{tmath|n}} to&nbsp;1 that have the same parity as <math display=inline>n.</math>

The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer <math display=inline>p,</math>

<math display=block>\begin{align}
   \operatorname{E}\left[|X - \mu|^p\right] &= \sigma^p (p-1)!! \cdot \begin{cases}
     \sqrt{\frac{2}{\pi}} & \text{if }p\text{ is odd} \\
     1 & \text{if }p\text{ is even}
   \end{cases} \\
   &= \sigma^p \cdot \frac{2^{p/2}\Gamma\left(\frac{p+1} 2 \right)}{\sqrt\pi}.
 \end{align}</math>
The last formula is valid also for any non-integer <math display=inline>p>-1.</math> When the mean <math display=inline>\mu \ne 0,</math> the plain and absolute moments can be expressed in terms of [[confluent hypergeometric function]]s <math display=inline>{}_1F_1</math> and <math display=inline>U.</math><ref>{{cite arXiv|last1=Winkelbauer|first1=Andreas|title=Moments and Absolute Moments of the Normal Distribution |date= 2012|class=math.ST |eprint=1209.4340}}</ref>

<math display="block">\begin{align}
   \operatorname{E}\left[X^p\right] &= \sigma^p\cdot {\left(-i\sqrt 2\right)}^p \, U{\left(-\frac{p}{2}, \frac{1}{2}, -\frac{\mu^2}{2\sigma^2}\right)}, \\
   \operatorname{E}\left[|X|^p \right] &= \sigma^p \cdot 2^{p/2} \frac {\Gamma{\left(\frac{1+p} 2\right)}}{\sqrt\pi} \, {}_1F_1{\left( -\frac{p}{2}, \frac{1}{2}, -\frac{\mu^2}{2\sigma^2} \right)}.
 \end{align}</math>

These expressions remain valid even if {{tmath|p}} is not an integer. See also [[Hermite polynomials#"Negative variance"|generalized Hermite polynomials]].

{| class="wikitable" style="margin: auto;"
|-
! Order !! Non-central moment, <math>\operatorname{E}\left[X^p\right]</math> !! Central moment, <math>\operatorname{E}\left[(X-\mu)^p\right]</math>
|-
| 1
| {{tmath|\mu}}
| {{tmath|0}}
|-
| 2
| <math display=inline>\mu^2+\sigma^2</math>
| <math display=inline>\sigma^2</math>
|-
| 3
| <math display=inline>\mu^3+3\mu\sigma^2</math>
| {{tmath|0}}
|-
| 4
| <math display=inline>\mu^4+6\mu^2\sigma^2+3\sigma^4</math>
| <math display=inline>3\sigma^4</math>
|-
| 5
| <math display=inline>\mu^5+10\mu^3\sigma^2+15\mu\sigma^4</math>
| {{tmath|0}}
|-
| 6
| <math display=inline>\mu^6+15\mu^4\sigma^2+45\mu^2\sigma^4+15\sigma^6</math>
| <math display=inline>15\sigma^6</math>
|-
| 7
| <math display=inline>\mu^7+21\mu^5\sigma^2+105\mu^3\sigma^4+105\mu\sigma^6</math>
| {{tmath|0}}
|-
| 8
| <math display=inline>\mu^8+28\mu^6\sigma^2+210\mu^4\sigma^4+420\mu^2\sigma^6+105\sigma^8</math>
| <math display=inline>105\sigma^8</math>
|}
The expectation of {{tmath|X}} conditioned on the event that {{tmath|X}} lies in an interval <math display=inline>[a,b]</math> is given by
<math display=block>\operatorname{E}\left[X \mid a<X<b \right] = \mu - \sigma^2\frac{f(b)-f(a)}{F(b)-F(a)}\,,</math>
where {{tmath|f}} and {{tmath|F}} respectively are the density and the cumulative distribution function of {{tmath|X}}. For <math display=inline>b=\infty</math> this is known as the [[inverse Mills ratio]]. Note that above, density {{tmath|f}} of {{tmath|X}} is used instead of standard normal density as in inverse Mills ratio, so here we have <math display=inline>\sigma^2</math> instead of {{tmath|\sigma}}.