The tpdf( Command

We're glad you came by, but you might find what you're looking for elsewhere. TI-Basic Developer is not the site it once was. While its information on commands and other calculator features remains almost second-to-none, its forum, archives, and even hosting service, Wikidot, have been decaying for years. The calculator community would love to see what you're working on, or help you in your next coding adventure, but TI-Basic Developer is no longer the place to do it. Instead, you should head over to Cemetech (primarily American) or TI-Planet (primarily international). Both are active, well-established forums with their own archives, chatrooms, reference material, and abundant coding tools and resources. We'll see you there, we hope.

Command Summary

Evaluates the Student's t probability density function with degrees of freedom ν.

Command Syntax

tpdf(t, ν)

Menu Location

Press:

1. 2ND DISTR to access the distribution menu
2. 4 to select tpdf(, or use arrows.

Press 5 instead of 4 on a TI-84+/SE with OS 2.30 or higher.

Calculator Compatibility

TI-83/84/+/SE

Token Size

2 bytes

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}\operatorname{tpdf}(x,\nu)=\operatorname{normalpdf}(x) \end{align}

Formulas

The value of tpdf( is given by

(2)

\begin{align} \operatorname{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} \operatorname{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

• tcdf(
• invT(
• Shade_t(

.