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# tinv

Student's t inverse cumulative distribution function

x = tinv(p,nu)

## Description

x = tinv(p,nu) returns the inverse of Student's t cdf using the degrees of freedom in nu for the corresponding probabilities in p. p and nu can be vectors, matrices, or multidimensional arrays that are the same size. A scalar input is expanded to a constant array with the same dimensions as the other inputs. The values in p must lie on the interval [0 1].

The t inverse function in terms of the t cdf is

$x={F}^{-1}\left(p|\nu \right)=\left\{x:F\left(x|\nu \right)=p\right\}$

where

$p=F\left(x|\nu \right)={\int }_{-\infty }^{x}\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}\frac{1}{\sqrt{\nu \pi }}\frac{1}{{\left(1+\frac{{t}^{2}}{\nu }\right)}^{\frac{\nu +1}{2}}}dt$

The result, x, is the solution of the cdf integral with parameter ν, where you supply the desired probability p.

## Examples

expand all

### Compute Student's t icdf

What is the 99th percentile of the Student's t distribution for one to six degrees of freedom?

```percentile = tinv(0.99,1:6)
```
```percentile =

31.8205    6.9646    4.5407    3.7469    3.3649    3.1427

```