Documentation

# gamcdf

Gamma cumulative distribution function

## Syntax

```gamcdf(x,a,b) [p,plo,pup] = gamcdf(x,a,b,pcov,alpha) [p,plo,pup] = gamcdf(___,'upper') ```

## Description

`gamcdf(x,a,b)` returns the gamma cdf at each of the values in `x` using the corresponding shape parameters in `a` and scale parameters in `b`. `x`, `a`, and `b` can be vectors, matrices, or multidimensional arrays that all have the same size. A scalar input is expanded to a constant array with the same dimensions as the other inputs. The parameters in `a` and `b` must be positive, and the values in `x` must lie on the interval `[0 Inf]`.

`[p,plo,pup] = gamcdf(x,a,b,pcov,alpha)` produces confidence bounds for `p` when the input parameters `a` and `b` are estimates. `pcov` is a 2-by-2 matrix containing the covariance matrix of the estimated parameters. `alpha` has a default value of 0.05, and specifies `100(1-alpha)`% confidence bounds. `plo` and `pup` are arrays of the same size as `p` containing the lower and upper confidence bounds.

`[p,plo,pup] = gamcdf(___,'upper')` returns the complement of the gamma cdf at each value in `x`, using an algorithm that more accurately computes the extreme upper tail probabilities. You can use the `'upper'` argument with any of the previous syntaxes.

The gamma cdf is

`$p=F\left(x|a,b\right)=\frac{1}{{b}^{a}\Gamma \left(a\right)}\underset{0}{\overset{x}{\int }}{t}^{a-1}{e}^{\frac{-t}{b}}dt$`

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

`gammainc` is the gamma distribution with b fixed at 1.

## Examples

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The mean of the gamma distribution is the product of the parameters, ab. In this example, the mean approaches the median as it increases (i.e., the distribution becomes more symmetric).

```a = 1:6; b = 5:10; prob = gamcdf(a.*b,a,b)```
```prob = 1×6 0.6321 0.5940 0.5768 0.5665 0.5595 0.5543 ```