# Documentation

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# icdf

Inverse cumulative distribution functions

## Syntax

``x = icdf('name',y,A)``
``x = icdf('name',y,A,B)``
``x = icdf('name',y,A,B,C)``
``x = icdf('name',y,A,B,C,D)``
``x = icdf(pd,y)``

## Description

example

````x = icdf('name',y,A)` returns the inverse cumulative distribution function (icdf) for the one-parameter distribution family specified by `'name'`, evaluated at the probability values in `y`. `A` contains the parameter value for the distribution.```

example

````x = icdf('name',y,A,B)` returns the icdf for the two-parameter distribution family specified by `'name'`, evaluated at the probability values in `y`. `A` and `B` contain the parameter values for the distribution.```
````x = icdf('name',y,A,B,C)` returns the icdf for the three-parameter distribution family specified by `'name'`, evaluated at the probability values in `y`. `A`, `B`, and `C` contain the parameter values for the distribution.```
````x = icdf('name',y,A,B,C,D)` returns the icdf for the four-parameter distribution family specified by `'name'`, evaluated at the probability values in `y`. `A`, `B`, `C`, and `D` contain the parameter values for the distribution.```

example

````x = icdf(pd,y)` returns the inverse cumulative distribution function of the probability distribution object, `pd`, evaluated at the probability values in `y`.```

## Examples

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Create a standard normal distribution object with the mean, , equal to 0 and the standard deviation, , equal to 1.

```mu = 0; sigma = 1; pd = makedist('Normal',mu,sigma); ```

Define the input vector y to contain the probability values at which to calculate the icdf.

```y = [0.1,0.25,0.5,0.75,0.9]; ```

Compute the icdf values for the standard normal distribution at the values in y.

```x = icdf(pd,y) ```
```x = -1.2816 -0.6745 0 0.6745 1.2816 ```

Each value in x corresponds to a value in the input vector y. For example, at the value y equal to 0.9, the corresponding icdf value x is equal to 1.2816.

Alternatively, you can compute the same icdf values without creating a probability distribution object. Use the `icdf` function and specify a standard normal distribution using the same parameter values for and .

```x2 = icdf('Normal',y,mu,sigma) ```
```x2 = -1.2816 -0.6745 0 0.6745 1.2816 ```

The icdf values are the same as those computed using the probability distribution object.

Create a Poisson distribution object with the rate parameter, , equal to 2.

```lambda = 2; pd = makedist('Poisson',lambda); ```

Define the input vector y to contain the probability values at which to calculate the icdf.

```y = [0.1,0.25,0.5,0.75,0.9]; ```

Compute the icdf values for the Poisson distribution at the values in y.

```x = icdf(pd,y) ```
```x = 0 1 2 3 4 ```

Each value in x corresponds to a value in the input vector y. For example, at the value y equal to 0.9, the corresponding icdf value x is equal to 4.

Alternatively, you can compute the same icdf values without creating a probability distribution object. Use the `icdf` function and specify a Poisson distribution using the same value for the rate parameter .

```x2 = icdf('Poisson',y,lambda) ```
```x2 = 0 1 2 3 4 ```

The icdf values are the same as those computed using the probability distribution object.

## Input Arguments

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Probability distribution name, specified as one of the following.

`name`DistributionInput Parameter AInput Parameter BInput Parameter CInput Parameter D
`'Beta'`Beta Distributiona: first shape parameterb: second shape parameter
`'Binomial'`Binomial Distributionn: number of trialsp: probability of success for each trial
`'BirnbaumSaunders'`Birnbaum-Saunders Distributionβ: scale parameterγ: shape parameter
`'Burr'`Burr Type XII Distributionα: scale parameterc: first shape parameterk: second shape parameter
`'Chisquare'`Chi-Square Distributionν: degrees of freedom
`'Exponential'`Exponential Distributionμ: mean
`'Extreme Value'`Extreme Value Distributionμ: location parameterσ: scale parameter
`'F'`F Distributionν1: numerator degrees of freedomν2: denominator degrees of freedom
`'Gamma'`Gamma Distributiona: shape parameterb: scale parameter
`'Generalized Extreme Value'`Generalized Extreme Value Distributionk: shape parameterσ: scale parameterμ: location parameter
`'Generalized Pareto'`Generalized Pareto Distributionk: tail index (shape) parameterσ: scale parameterμ: threshold (location) parameter
`'Geometric'`Geometric Distributionp: probability parameter
`'HalfNormal'`Half-Normal Distributionμ: location parameterσ: scale parameter
`'Hypergeometric'`Hypergeometric Distributionm: size of the populationk: number of items with the desired characteristic in the populationn: number of samples drawn
`'InverseGaussian'`Inverse Gaussian Distributionμ: scale parameterλ: shape parameter
`'Logistic'`Logistic Distributionμ: meanσ: scale parameter
`'LogLogistic'`Loglogistic Distributionμ: log meanσ: log scale parameter
`'Lognormal'`Lognormal Distributionμ: log meanσ: log standard deviation
`'Nakagami'`Nakagami Distributionμ: shape parameterω: scale parameter
`'Negative Binomial'`Negative Binomial Distributionr: number of successesp: probability of success in a single trial
`'Noncentral F'`Noncentral F Distributionν1: numerator degrees of freedomν2: denominator degrees of freedomδ: noncentrality parameter
`'Noncentral t'`Noncentral t Distributionν: degrees of freedomδ: noncentrality parameter
`'Noncentral Chi-square'`Noncentral Chi-Square Distributionν: degrees of freedomδ: noncentrality parameter
`'Normal'`Normal Distributionμ: mean σ: standard deviation
`'Poisson'`Poisson Distributionλ: mean
`'Rayleigh'`Rayleigh Distributionb: scale parameter
`'Rician'`Rician Distributions: noncentrality parameterσ: scale parameter
`'Stable'`Stable Distributionα: first shape parameterβ: second shape parameterγ: scale parameterδ: location parameter
`'T'`Student's t Distributionν: degrees of freedom
`'tLocationScale'`t Location-Scale Distributionμ: location parameterσ: scale parameterν: shape parameter
`'Uniform'`Uniform Distribution (Continuous)a: lower endpoint (minimum)b: upper endpoint (maximum)
`'Discrete Uniform'`Uniform Distribution (Discrete)n: maximum observable value
`'Weibull'`Weibull Distributiona: scale parameterb: shape parameter

Probability values at which to evaluate the icdf, specified as a scalar value, or an array of scalar values.

• If `x` is a scalar value, and if you specify distribution parameters `A`, `B`, `C`, or `D` as arrays, then `cdf` expands `x` into a constant array of the same size as the parameters.

• If `x` is an array, and if you specify distribution parameters `A`, `B`, `C`, or `D` as arrays, then `x`, `A`, `B`, `C`, and `D` must all be the same size.

Example: `[0.1,0.25,0.5,0.75,0.9]`

Data Types: `single` | `double`

First probability distribution parameter, specified as a scalar value, or an array of scalar values.

If `x` and `A` are arrays, they must be the same size. If `x` is a scalar, then `cdf` expands it into a constant matrix the same size as `A`. If `A` is a scalar, then `cdf` expands it into a constant matrix the same size as `x`.

Data Types: `single` | `double`

Second probability distribution parameter, specified as a scalar value, or an array of scalar values.

If `x`, `A`, and `B` are arrays, they must be the same size. If `x` is a scalar, then `cdf` expands it into a constant matrix the same size as `A` and `B`. If `A` or `B` are scalars, then `cdf` expands them into constant matrices the same size as `x`

Data Types: `single` | `double`

Third probability distribution parameter, specified as a scalar value, or an array of scalar values.

If `x`, `A`, `B`, and `C` are arrays, they must be the same size. If `x` is a scalar, then `cdf` expands it into a constant matrix the same size as `A`, `B`, and `C`. If any of `A`, `B` or `C` are scalars, then `cdf` expands them into constant matrices the same size as `x`.

Data Types: `single` | `double`

Fourth probability distribution parameter, specified as a scalar value, or an array of scalar values.

If `x`, `A`, `B`, `C`, and `D` are arrays, they must be the same size. If `x` is a scalar, then `cdf` expands it into a constant array the same size as `A`, `B`, `C`, and `D`. If any of `A`, `B` , `C`, or `D` are scalars, then `cdf` expands them into constant matrices the same size as `x`.

Data Types: `single` | `double`

Probability distribution, specified as a probability distribution object created using one of the following.

 `makedist` Create a probability distribution object using specified parameter values. `fitdist` Fit a probability distribution object to sample data. `distributionFitter` Fit a probability distribution object to sample data using the interactive Distribution Fitting app. `paretotails` Create a Pareto tails object.

## Output Arguments

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Inverse cumulative distribution function of the specified probability distribution, returned as an array.