# fcdf

F cumulative distribution function

## Syntax

```p = fcdf(x,v1,v2) p = fcdf(x,v1,v2,'upper') ```

## Description

`p = fcdf(x,v1,v2)` computes the F cdf at each of the values in `x` using the corresponding numerator degrees of freedom `v1` and denominator degrees of freedom `v2`. `x`, `v1`, and `v2` can be vectors, matrices, or multidimensional arrays that are all the same size. A scalar input is expanded to a constant matrix with the same dimensions as the other inputs. `v1` and `v2` parameters must contain real positive values, and the values in `x` must lie on the interval `[0 Inf]`.

`p = fcdf(x,v1,v2,'upper')` returns the complement of the F cdf at each value in `x`, using an algorithm that more accurately computes the extreme upper tail probabilities.

The F cdf is

`$p=F\left(x|{\nu }_{1},{\nu }_{2}\right)={\int }_{0}^{x}\frac{\Gamma \left[\frac{\left({\nu }_{1}+{\nu }_{2}\right)}{2}\right]}{\Gamma \left(\frac{{\nu }_{1}}{2}\right)\Gamma \left(\frac{{\nu }_{2}}{2}\right)}{\left(\frac{{\nu }_{1}}{{\nu }_{2}}\right)}^{\frac{{\nu }_{1}}{2}}\frac{{t}^{\frac{{\nu }_{1}-2}{2}}}{{\left[1+\left(\frac{{\nu }_{1}}{{\nu }_{2}}\right)t\right]}^{\frac{{\nu }_{1}+{\nu }_{2}}{2}}}dt$`

The result, p, is the probability that a single observation from an F distribution with parameters ν1 and ν2 will fall in the interval [0 x].

## Examples

collapse all

The following illustrates a useful mathematical identity for the F distribution.

```nu1 = 1:5; nu2 = 6:10; x = 2:6; F1 = fcdf(x,nu1,nu2)```
```F1 = 1×5 0.7930 0.8854 0.9481 0.9788 0.9919 ```
`F2 = 1 - fcdf(1./x,nu2,nu1)`
```F2 = 1×5 0.7930 0.8854 0.9481 0.9788 0.9919 ```