# chi2pdf

Chi-square probability density function

## Syntax

``y = chi2pdf(x,nu)``

## Description

example

````y = chi2pdf(x,nu)` returns the probability density function (pdf) of the chi-square distribution with `nu` degrees of freedom, evaluated at the values in `x`.```

## Examples

collapse all

Compute the density of the observed value 2 in the chi-square distribution with `3` degrees of freedom.

`y1 = chi2pdf(2,3)`
```y1 = 0.2076 ```

Compute the density of the observed value `4` in the chi-square distributions with degrees of freedom `1` through 6.

`y2 = chi2pdf(4,1:6)`
```y2 = 1×6 0.0270 0.0677 0.1080 0.1353 0.1440 0.1353 ```

The mean of the chi-square distribution is equal to the degrees of freedom. Compute the density of the mean for the chi-square distributions with degrees of freedom `1` through `6`.

```nu = 1:6; x = nu; y3 = chi2pdf(x,nu)```
```y3 = 1×6 0.2420 0.1839 0.1542 0.1353 0.1220 0.1120 ```

As the degrees of freedom increase, the density of the mean decreases.

## Input Arguments

collapse all

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 `nu` using an array.

If either or both of the input arguments `x` and `nu` are arrays, then the array sizes must be the same. In this case, `chi2pdf` 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 element in `nu`, evaluated at the corresponding element in `x`.

Example: `[3 4 7 9]`

Data Types: `single` | `double`

Degrees of freedom for the chi-square 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 `nu` using an array.

If either or both of the input arguments `x` and `nu` are arrays, then the array sizes must be the same. In this case, `chi2pdf` 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 element in `nu`, evaluated at the corresponding element in `x`.

Example: `[9 19 49 99]`

Data Types: `single` | `double`

## Output Arguments

collapse all

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

collapse all

### Chi-Square pdf

The chi-square distribution is a one-parameter family of curves. The parameter ν is the degrees of freedom.

The pdf of the chi-square distribution is

`$y=f\left(x|\nu \right)=\frac{{x}^{\left(\nu -2\right)/2}{e}^{-x/2}}{{2}^{\frac{\nu }{2}}\Gamma \left(\nu /2\right)},$`

where ν is the degrees of freedom and Γ( · ) is the Gamma function.

## Alternative Functionality

• `chi2pdf` is a function specific to the chi-square distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `pdf`, which supports various probability distributions. To use `pdf`, specify the probability distribution name and its parameters. Note that the distribution-specific function `chi2pdf` 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.