The Student-t distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The StudentT variate is related to the Normal variate and the ChiSquare variate by the formula StudentT(nu) ~ Normal(0,1)/sqrt(ChiSquare(nu)/nu)

The StudentT variate with degrees of freedom 1 is related to the standard Cauchy variate by StudentT(1) ~ Cauchy(0,1).

Note that the StudentT command is inert and should be used in combination with the RandomVariable command.

with(Statistics):

X := RandomVariable(StudentT(nu)):

PDF(X, u);

$\frac{\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)}{\sqrt{\mathrm{\pi }\mathrm{\nu }}\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right){\left(1+\frac{u^2}{\mathrm{\nu }}\right)}^{\frac{\mathrm{\nu }}{2}+\frac{1}{2}}}$

PDF(X, .5);

$\frac{0.5641895835\mathrm{\Gamma }\left(0.5000000000\mathrm{\nu }+0.5000000000\right)}{\sqrt{\mathrm{\nu }}\mathrm{\Gamma }\left(0.5000000000\mathrm{\nu }\right){\left(1.+\frac{0.25}{\mathrm{\nu }}\right)}^{0.5000000000\mathrm{\nu }+0.5000000000}}$

Mean(X);

$\left\{\begin{array}{cc}\mathrm{undefined}& \mathrm{\nu }\le 1\\ 0& \mathrm{otherwise}\end{array}\right.$

Variance(X);

$\left\{\begin{array}{cc}\mathrm{undefined}& \mathrm{\nu }\le 2\\ \frac{\mathrm{\nu }}{-2+\mathrm{\nu }}& \mathrm{otherwise}\end{array}\right.$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.