# gampdf

Gamma probability density function

## Syntax

``y = gampdf(x,a)``
``y = gampdf(x,a,b)``

## Description

example

````y = gampdf(x,a)` returns the probability density function (pdf) of the standard gamma distribution with the shape parameter `a`, evaluated at the values in `x`.```

example

````y = gampdf(x,a,b)` returns the pdf of the gamma distribution with the shape parameter `a` and the scale parameter `b`, evaluated at the values in `x`.```

## Examples

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Compute the density of the observed value `5` in the standard gamma distribution with shape parameter `2`.

`y1 = gampdf(5,2)`
```y1 = 0.0337 ```

Compute the density of the observed value `5` in the gamma distributions with shape parameter `2` and scale parameters `1` through `5`.

`y2 = gampdf(5,2,1:5)`
```y2 = 1×5 0.0337 0.1026 0.1049 0.0895 0.0736 ```

## Input Arguments

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Values at which to evaluate the pdf, specified as a nonnegative scalar value or an array of nonnegative scalar values.

• To evaluate the pdf at multiple values, specify `x` using an array.

• To evaluate the pdfs of multiple distributions, specify `a` and `b` using arrays.

If one or more of the input arguments `x`, `a`, and `b` are arrays, then the array sizes must be the same. In this case, `gampdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `y` is the pdf value of the distribution specified by the corresponding elements in `a` and `b`, evaluated at the corresponding element in `x`.

Example: `[3 4 7 9]`

Data Types: `single` | `double`

Shape parameter of the gamma distribution, specified as a positive scalar value or an array of positive scalar values.

• To evaluate the pdf at multiple values, specify `x` using an array.

• To evaluate the pdfs of multiple distributions, specify `a` and `b` using arrays.

If one or more of the input arguments `x`, `a`, and `b` are arrays, then the array sizes must be the same. In this case, `gampdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `y` is the pdf value of the distribution specified by the corresponding elements in `a` and `b`, evaluated at the corresponding element in `x`.

Example: `[1 2 3 5]`

Data Types: `single` | `double`

Scale parameter of the gamma distribution, specified as a positive scalar value or an array of positive scalar values.

• To evaluate the pdf at multiple values, specify `x` using an array.

• To evaluate the pdfs of multiple distributions, specify `a` and `b` using arrays.

If one or more of the input arguments `x`, `a`, and `b` are arrays, then the array sizes must be the same. In this case, `gampdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `y` is the pdf value of the distribution specified by the corresponding elements in `a` and `b`, evaluated at the corresponding element in `x`.

Example: `[1 1 2 2]`

Data Types: `single` | `double`

## Output Arguments

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pdf values evaluated at the values in `x`, returned as a scalar value or an array of scalar values. `y` is the same size as `x`, `a`, and `b` after any necessary scalar expansion. Each element in `y` is the pdf value of the distribution specified by the corresponding elements in `a` and `b`, evaluated at the corresponding element in `x`.

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### Gamma pdf

The gamma distribution is a two-parameter family of curves. The parameters a and b are shape and scale, respectively.

The gamma pdf is

`$y=f\left(x|a,b\right)=\frac{1}{{b}^{a}\Gamma \left(a\right)}{x}^{a-1}{e}^{\frac{-x}{b}},$`

where Γ( · ) is the Gamma function.

The standard gamma distribution occurs when b = 1.

## Alternative Functionality

• `gampdf` is a function specific to the gamma distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `pdf`, which supports various probability distributions. To use `pdf`, create a `GammaDistribution` probability distribution object and pass the object as an input argument or specify the probability distribution name and its parameters. Note that the distribution-specific function `gampdf` is faster than the generic function `pdf`.

• Use the Probability Distribution Function app to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution.

## Version History

Introduced before R2006a