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Student's z-Distribution

The probability density function and cumulative distribution functions for Student's $z$-distribution are given by

$\displaystyle f(z)$ $\textstyle =$ $\displaystyle {\Gamma\left({{\textstyle{n\over 2}}}\right)\over \sqrt{\pi}\, \Gamma\left({{\textstyle{n-1\over 2}}}\right)} (1+z^2)^{-n/2}$ (1)
$\displaystyle D(z)$ $\textstyle =$ $\displaystyle {-z^{1-n}\Gamma({\textstyle{1\over 2}}n)\,{}_2F_1({\textstyle{1\o... ...over 2}}(n+1); -z^{-2})\over 2\sqrt{\pi}\,\Gamma[{\textstyle{1\over 2}}(n+1)]}.$  
      (2)

The Mean is 0, so the Moments are
$\displaystyle \mu_1$ $\textstyle =$ $\displaystyle 0$ (3)
$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle {1\over n-3}$ (4)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle 0$ (5)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle {3\over (n-3)(n-5)}.$ (6)

The Mean, Variance, Skewness, and Kurtosis are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle 0$ (7)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle {1\over n-3}$ (8)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle 0$ (9)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle {6\over n-5}.$ (10)

Letting
\begin{displaymath} z\equiv {(\bar x-\mu)\over s}, \end{displaymath} (11)

where $x$ is the sample Mean and $\mu$ is the population Mean gives Student's t-Distribution.

See also Student's t-Distribution



© 1996-9 Eric W. Weisstein
1999-05-26