# chi2cdf

Chi-square cumulative distribution function

## Syntax

``p = chi2cdf(x,nu)``
``p = chi2cdf(x,nu,'upper')``

## Description

example

````p = chi2cdf(x,nu)` returns the cumulative distribution function (cdf) of the chi-square distribution with degrees of freedom `nu`, evaluated at the values in `x`.```

example

````p = chi2cdf(x,nu,'upper')` returns the complement of the cdf, evaluated at the values in `x` with degrees of freedom `nu`, using an algorithm that more accurately computes the extreme upper-tail probabilities than subtracting the lower tail value from 1.```

## Examples

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Compute the probability that an observation from the chi-square distribution with `5` degrees of freedom is in the interval `[0 3]`.

`p1 = chi2cdf(3,5)`
```p1 = 0.3000 ```

Compute the probability that an observation from the chi-square distributions with degrees of freedom `1` through `5` is in the interval `[0 3]`.

`p2 = chi2cdf(3,1:5)`
```p2 = 1×5 0.9167 0.7769 0.6084 0.4422 0.3000 ```

The mean of the chi-square distribution is equal to the degrees of freedom. Compute the probability that an observation is in the interval `[0 nu]` for degrees of freedom `1` through `6`.

```nu = 1:6; x = nu; p3 = chi2cdf(x,nu)```
```p3 = 1×6 0.6827 0.6321 0.6084 0.5940 0.5841 0.5768 ```

As the degrees of freedom increase, the probability that an observation from a chi-square distribution with degrees of freedom `nu` is less than the mean value approaches `0.5`.

Determine the probability that an observation from the chi-square distribution with `3` degrees of freedom is in on the interval `[100 Inf]`.

`p1 = 1 - chi2cdf(100,3)`
```p1 = 0 ```

`chi2cdf(100,3)` is nearly `1`, so `p1` becomes `0`. Specify `'upper'` so that `chi2cdf` computes the extreme upper-tail probabilities more accurately.

`p2 = chi2cdf(100,3,'upper')`
```p2 = 1.5542e-21 ```

## Input Arguments

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

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

• To evaluate the cdfs 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, `chi2cdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf 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 cdf at multiple values, specify `x` using an array.

• To evaluate the cdfs 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, `chi2cdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf 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

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cdf 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 `p` is the cdf value of the distribution specified by the corresponding element in `nu`, evaluated at the corresponding element in `x`.

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### Chi-Square cdf

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

The cdf of the chi-square distribution is

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

where ν is the degrees of freedom and Γ( · ) is the Gamma function. The result p is the probability that a single observation from the chi-square distribution with ν degrees of freedom falls in the interval [0, x].

## Alternative Functionality

• `chi2cdf` is a function specific to the chi-square distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `cdf`, which supports various probability distributions. To use `cdf`, specify the probability distribution name and its parameters. Note that the distribution-specific function `chi2cdf` is faster than the generic function `cdf`.

• 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.