| Property | Value |
|---|---|
| Hex Value | $BB1C |
| Categories | |
| Localizations |
|
tpdf(
Overview
Computes the probability density function (pdf) for the Student-t distribution at a specified x value with specified degrees of freedom df.
Availability: Token available everywhere.
Syntax
tpdf(x,df)
Arguments
| Name | Type | Optional |
|---|---|---|
| x | ||
| df |
Location
2nd, distr, DISTR, 5:tpdf(
Description
tpdf( is the Student's t probability density function.
Since the t distribution is continuous, the value of tpdf( doesn't represent an actual probability — in fact, one of the few uses for this command is to draw a graph of the bell curve. You could also use it for various calculus purposes, such as finding inflection points.
The command takes two arguments: the first is the value where the PDF is to be evaluated, and the second is the number of degrees of freedom (so the calculator knows which t distribution to use). As the degrees of freedom increases without bound, tpdf( approaches normalpdf(; i.e.
(1) \(\begin{align} \lim_{\nu\rightarrow\infty}\texttt{tpdf}(x,\nu)=\texttt{normalpdf}(x) \end{align}\)
Formulas
The value of tpdf( is given by
(2) \(\begin{align} \texttt{tpdf}(t,\nu) = \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\,\Gamma(\nu/2)}\,\left(1+\frac{t^2}{\nu}\right)^{-\frac1{2}(\nu+1)} \end{align}\)
(where Γ is the gamma function), or alternatively
(3) \(\begin{align} \texttt{tpdf}(t,\nu) = \frac1{\sqrt{\nu}B(\nu/2,1/2)}\,\left(1+\frac{t^2}{\nu}\right)^{-\frac1{2}(\nu+1)} \end{align}\)
(where B is the beta function)
Related Commands
Source: parts of this page were written by the following TI|BD contributors: burr, DarkerLine, GoVegan, kg583, thornahawk.
History
| Calculator | OS Version | Description |
|---|---|---|
| TI-83 | 0.01013 | Added |
Created: February 23, 2023 23:15:01