Editing Skewness (section)

===Fisher's moment coefficient of skewness===
The skewness <math>\gamma_1</math> of a random variable {{mvar|X}} is the third [[standardized moment]] <math>\tilde{\mu}_3</math>, defined as:<ref name="StanBrown1"/><ref name="FXSolver1"/>

<math display="block">
    \gamma_1 := \tilde{\mu}_3 = \operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3 \right]
             = \frac{\mu_3}{\sigma^3}
             = \frac{\operatorname{E}\left[(X-\mu)^3\right]}{\left( \operatorname{E}\left[ (X-\mu)^2 \right] \right)^{3/2}}
             = \frac{\kappa_3}{\kappa_2^{3/2}}
  </math>
where {{mvar|μ}} is the mean, {{mvar|σ}} is the [[standard deviation]], E is the [[expected value|expectation operator]], {{math|''μ''<sub>3</sub>}} is the third [[central moment]], and {{math|''κ''<sub>''t''</sub>}} are the {{mvar|t}}-th [[cumulant]]s. It is sometimes referred to as '''Pearson's moment coefficient of skewness''',<ref name="FXSolver1">[http://www.fxsolver.com/browse/formulas/Pearson's+moment+coefficient+of+skewness Pearson's moment coefficient of skewness], FXSolver.com</ref> or simply the '''moment coefficient of skewness''',<ref name="StanBrown1">[http://brownmath.com/stat/shape.htm "Measures of Shape: Skewness and Kurtosis"], 2008–2016 by Stan Brown, Oak Road Systems</ref> but should not be confused with Pearson's other skewness statistics (see below). The last equality expresses skewness in terms of the ratio of the third cumulant {{math|''κ''<sub>3</sub>}} to the 1.5th power of the second cumulant {{math|''κ''<sub>2</sub>}}. This is analogous to the definition of [[kurtosis]] as the fourth cumulant normalized by the square of the second cumulant. 
The skewness is also sometimes denoted {{math|Skew[''X'']}}.

If {{mvar|σ}} is finite and {{mvar|μ}} is finite too, then skewness can be expressed in terms of the non-central moment {{math|E[''X''<sup>3</sup>]}} by expanding the previous formula:
<math display="block">
\begin{align}
    \tilde{\mu}_3
     &= \operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3 \right] \\
     &= \frac{\operatorname{E}[X^3] - 3\mu\operatorname E[X^2] + 3\mu^2\operatorname E[X] - \mu^3}{\sigma^3}\\
     &= \frac{\operatorname{E}[X^3] - 3\mu(\operatorname E[X^2] -\mu\operatorname E[X]) - \mu^3}{\sigma^3}\\
     &= \frac{\operatorname{E}[X^3] - 3\mu\sigma^2 - \mu^3}{\sigma^3}.
\end{align}
  </math>
<!-- EDITORS BEWARE: DO NOT CHANGE THIS INTO   E[X^3] - 3\mu E[X^2] + 2\mu^3  /// SEE TALK PAGE -->