The gamma function
Γ(t)
that appears
in the definition of the Student's t
distribution (and the gamma distribution) is the continuous function
Γt
=
e-xxt-1dx defined for positive
real numbers t. The gamma function satisfies
The Student's t cumulative distribution function
TDist(x;v) is defined by the integral
The function
TInv(p;v) is the value of x for which the
integral has the value p, as demonstrated here:
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The following plots display the density and distribution functions
TDen
x;v
and
TDist
x;v
for the
parameters v = 1 and v = 15 with
-5≤x≤5.
Note that the Student's t density functions resemble the standard normal
density function in shape, although these curves are a bit flatter at the
center. It is not difficult to show, using Scientific Notebook and
the definitions of the two density functions, that
TDen(u;v) =
NormalDen(u), the density function
for the standard normal distribution.
Student's t distribution tables list values of the inverse distribution function corresponding to probabilities (values of the distribution function) and degrees of freedom. For values of v above 30, the normal distribution is such a close approximation for the Student's t distribution that tables usually provide values only up to v = 30.