The tpdf( Command

Command Summary

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

Command Syntax

tpdf(t, ν)

Menu Location


  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


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.

\begin{align} \definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen} \lim_{\nu\rightarrow\infty}\operatorname{tpdf}(x,\nu)=\operatorname{normalpdf}(x) \end{align}


The value of tpdf( is given by

\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

\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)

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