The student's t distribution has density function (for n = 1, 2, ...):

f(x) = G[(n+1)/2]/

where

n |
is the degrees of freedom |

G |
(gamma) is the Gamma function |

p |
is the constant Pi (3.14...) |

The animation above shows how the shape of the density and distribution functions change as the degrees of freedom increase. Note that optimum scaling (rather than fixed scaling) is used in the animation (see Probability Distribution Calculator for further details).

For a complete listing of all distribution functions, see Distributions and Their Functions.