# TI-BASIC:Tpdf

**Command Summary**

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

**Command Syntax**

tpdf(*t*, *ν*)

**Menu Location**

Press:

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

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

TI-83/84/+/SE

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.

<math> \definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen} \lim_{\nu\rightarrow\infty}\operatorname{tpdf}(x,\nu)=\operatorname{normalpdf}(x) </math>

# Formulas

The value of tpdf( is given by

(where Γ is the [wikipedia:Gamma_function gamma function]), or alternatively

(where B is the [wikipedia:Beta_function beta function])