Student's t

The gamma function Γ(t) that appears in the definition of the Student's t distribution (and the gamma distribution) is the continuous function Γ$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = $\int_{{0}}^{{\infty }}$e-xxt-1dx defined for positive real numbers t. The gamma function satisfies

Γ(1) =  1 and Γ(t + 1) = (t)

and for positive integers k, it is the familiar factorial function

Γ(k) = $\displaystyle \left(\vphantom{ k-1}\right.$k - 1$\displaystyle \left.\vphantom{ k-1}\right)$!

The gamma functionDM6-4.tex#Gamma function is active in Scientific Notebook. For example, place the insertion point in the expression Γ(5) and choose Evaluate to get Γ(5) =  24. Note that 24 = 4×3×2×1.

The Student's t cumulative distribution function $\limfunc$TDist(x;v) is defined by the integral

$\displaystyle \limfunc$TDist(x;v) = $\displaystyle {\frac{{\Gamma (\frac{v+1}{2})}}{{\Gamma (\frac{v}{2})\sqrt{%
\pi v}}}}$$\displaystyle \int_{{-\infty }}^{{x}}$$\displaystyle \left(\vphantom{ 1+\frac{1}{v}u^{2}}\right.$1 + $\displaystyle {\frac{{1}}{{v}}}$u2$\displaystyle \left.\vphantom{ 1+\frac{1}{v}u^{2}}\right)^{{-\frac{v+1}{2}%
}}_{}$du

of the density function

$\displaystyle \limfunc$TDen(u;v) = $\displaystyle {\frac{{\Gamma (\frac{v+1}{2})}}{{\Gamma (\frac{v}{2})\sqrt{%
\pi v}}}}$$\displaystyle \left(\vphantom{ 1+\frac{1}{v}u^{2}}\right.$1 + $\displaystyle {\frac{{1}}{{v}}}$u2$\displaystyle \left.\vphantom{ 1+\frac{1}{v}u^{2}}\right)^{{-\frac{v+1}{2}}}_{}$

with shape parameter v, called degrees of freedom, that ranges over the positive integers. The variance for a Student's t distribution is ${\frac{{v}}{{%
v-2}}}$, provided v > 2.

The function $\limfunc$TInv(p;v) is the value of x for which the integral has the value p, as demonstrated here:

$\displaystyle \limfunc$TDist(63.66;1) =  .995   $\displaystyle \limfunc$TDist(- .97847;3) =  .2
$\displaystyle \limfunc$TInv(.995;1) =  63.66   $\displaystyle \limfunc$TInv(.2;3) =   - .97847

The following plots display the density and distribution functions $\func$TDen$\left(\vphantom{ x;v}\right.$x;v$\left.\vphantom{ x;v}\right)$ and $\func$TDist$\left(\vphantom{ x;v}\right.$x;v$\left.\vphantom{ x;v}\right)$ for the parameters v = 1 and v = 15  with -5≤x≤5.


itbpFU2.1577in1.7573in0inStudent's t density functions itbpFU2.1577in1.7573in0inStudent's t distribution functions

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 $\lim_{{v\rightarrow
\infty }}^{}$$\limfunc$TDen(u;v) = $\limfunc$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.



\begin{example}
Assuming a Student's t distribution with $5$\ degrees of freedo...
...math}
\limfunc{TInv}(.95;5)=2.015
\end{displaymath}
\medskip
\end{example}