Editing Student's t-distribution (section)

===Moments===
For <math>\nu > 1\ ,</math> the [[raw moment]]s of the {{mvar|t}}&nbsp;distribution are

:<math>\operatorname{\mathbb E}\left\{\ T^k\ \right\} = \begin{cases}
\quad 0 & k \text{ odd }, \quad 0 < k < \nu\ , \\ {} \\
\frac{1}{\ \sqrt{\pi\ }\ \Gamma\left(\frac{\ \nu\ }{ 2 }\right)}\ \left[\ \Gamma\!\left(\frac{\ k + 1\ }{ 2 }\right)\ \Gamma\!\left(\frac{\ \nu - k\ }{ 2 }\right)\ \nu^{\frac{\ k\ }{ 2 }}\ \right] & k \text{ even }, \quad 0 < k < \nu ~.\\
\end{cases}</math>

Moments of order <math>\ \nu\ </math> or higher do not exist.<ref>{{cite book |vauthors=Casella G, Berger RL |year=1990 |title=Statistical Inference |publisher=Duxbury Resource Center |isbn=9780534119584 |page =56}}</ref>

The term for <math>\ 0 < k < \nu\ ,</math> {{mvar|k}} even, may be simplified using the properties of the [[gamma function]] to

:<math>\operatorname{\mathbb E}\left\{\ T^k\ \right\} = \nu^{ \frac{\ k\ }{ 2 } }\ \prod_{j=1}^{k/2}\ \frac{~ 2j - 1 ~}{ \nu - 2j } \qquad k \text{ even}, \quad 0 < k < \nu ~.</math>

For a {{mvar|t}}&nbsp;distribution with <math>\ \nu\ </math> degrees of freedom, the [[expected value]] is <math>\ 0\ </math> if <math>\ \nu > 1\ ,</math> and its [[variance]] is <math>\ \frac{ \nu }{\ \nu-2\ }\ </math> if <math>\ \nu > 2 ~.</math> The [[skewness]] is 0 if <math>\ \nu > 3\ </math> and the [[excess kurtosis]] is <math>\ \frac{ 6 }{\ \nu - 4\ }\ </math> if <math>\ \nu > 4 ~.</math>