Documentation

# tinv

Student's t inverse cumulative distribution function

## Syntax

```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

collapse all

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 = 1×6 31.8205 6.9646 4.5407 3.7469 3.3649 3.1427 ```