Student's t-distribution
Moments

For ν > 1 , {\displaystyle \nu >1\,} the raw moments of the t distribution are

Moments of order ν {\displaystyle \ \nu \ } or higher do not exist.

The term for 0 < k < ν , {\displaystyle \ 0<k<\nu \,} k even, may be simplified using the properties of the gamma function to

For a t distribution with ν {\displaystyle \ \nu \ } degrees of freedom, the expected value is 0 {\displaystyle \ 0\ } if ν > 1 , {\displaystyle \ \nu >1\,} and its variance is ν ν − 2 {\displaystyle \ {\frac {\nu }{\ \nu -2\ }}\ } if ν > 2. {\displaystyle \ \nu >2~.} The skewness is 0 if ν > 3 {\displaystyle \ \nu >3\ } and the excess kurtosis is 6 ν − 4 {\displaystyle \ {\frac {6}{\ \nu -4\ }}\ } if ν > 4. {\displaystyle \ \nu >4~.}
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