Documentation

# cdf

Cumulative distribution function

## Syntax

``y = cdf('name',x,A)``
``y = cdf('name',x,A,B)``
``y = cdf('name',x,A,B,C)``
``y = cdf('name',x,A,B,C,D)``
``y = cdf(pd,x)``
``y = cdf(___,'upper')``

## Description

example

````y = cdf('name',x,A)` returns the cumulative distribution function (cdf) for the one-parameter distribution family specified by `'name'` and the distribution parameter `A`, evaluated at the values in `x`.```

example

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

example

````y = cdf(pd,x)` returns the cdf of the probability distribution object `pd`, evaluated at the values in `x`.```
````y = cdf(___,'upper')` returns the complement of the cdf using an algorithm that more accurately computes the extreme upper-tail probabilities. `'upper'` can follow any of the input arguments in the previous syntaxes.```

## Examples

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

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

Define the input vector x to contain the values at which to calculate the cdf.

`x = [-2,-1,0,1,2];`

Compute the cdf values for the standard normal distribution at the values in x.

`y = cdf(pd,x)`
```y = 1×5 0.0228 0.1587 0.5000 0.8413 0.9772 ```

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 1, the corresponding cdf value y is equal to 0.8413.

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

`y2 = cdf('Normal',x,mu,sigma)`
```y2 = 1×5 0.0228 0.1587 0.5000 0.8413 0.9772 ```

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

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

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

Define the input vector x to contain the values at which to calculate the cdf.

`x = [0,1,2,3,4];`

Compute the cdf values for the Poisson distribution at the values in x.

`y = cdf(pd,x)`
```y = 1×5 0.1353 0.4060 0.6767 0.8571 0.9473 ```

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 3, the corresponding cdf value y is equal to 0.8571.

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

`y2 = cdf('Poisson',x,lambda)`
```y2 = 1×5 0.1353 0.4060 0.6767 0.8571 0.9473 ```

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

Create a standard normal distribution object.

`pd = makedist('Normal')`
```pd = NormalDistribution Normal distribution mu = 0 sigma = 1 ```

Specify the `x` values and compute the cdf.

```x = -3:.1:3; p = cdf(pd,x);```

Plot the cdf of the standard normal distribution.

`plot(x,p)`

Create three gamma distribution objects. The first uses the default parameter values. The second specifies `a = 1` and `b = 2`. The third specifies `a = 2` and `b = 1`.

`pd_gamma = makedist('Gamma')`
```pd_gamma = GammaDistribution Gamma distribution a = 1 b = 1 ```
`pd_12 = makedist('Gamma','a',1,'b',2)`
```pd_12 = GammaDistribution Gamma distribution a = 1 b = 2 ```
`pd_21 = makedist('Gamma','a',2,'b',1)`
```pd_21 = GammaDistribution Gamma distribution a = 2 b = 1 ```

Specify the `x` values and compute the cdf for each distribution.

```x = 0:.1:5; cdf_gamma = cdf(pd_gamma,x); cdf_12 = cdf(pd_12,x); cdf_21 = cdf(pd_21,x);```

Create a plot to visualize how the cdf of the gamma distribution changes when you specify different values for the shape parameters `a` and `b`.

```figure; J = plot(x,cdf_gamma); hold on; K = plot(x,cdf_12,'r--'); L = plot(x,cdf_21,'k-.'); set(J,'LineWidth',2); set(K,'LineWidth',2); legend([J K L],'a = 1, b = 1','a = 1, b = 2','a = 2, b = 1','Location','southeast'); hold off;```

Fit Pareto tails to a $t$ distribution at cumulative probabilities 0.1 and 0.9.

```t = trnd(3,100,1); obj = paretotails(t,0.1,0.9); [p,q] = boundary(obj)```
```p = 2×1 0.1000 0.9000 ```
```q = 2×1 -1.8487 2.0766 ```

Compute the cdf at the values in `q`.

`cdf(obj,q)`
```ans = 2×1 0.1000 0.9000 ```

## Input Arguments

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Probability distribution name, specified as one of the probability distribution names in this table.

`'name'`DistributionInput Parameter `A`Input Parameter `B`Input Parameter `C`Input 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 mean of logarithmic valuesσ scale parameter of logarithmic values
`'Lognormal'`Lognormal Distributionμ mean of logarithmic valuesσ standard deviation of logarithmic values
`'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

Example: `'Normal'`

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

If one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.

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 one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.

Data Types: `single` | `double`

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

If one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.

Data Types: `single` | `double`

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

If one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.

Data Types: `single` | `double`

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

If one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.

Data Types: `single` | `double`

Probability distribution, specified as a probability distribution object created with a function or app in this table.

Function or AppDescription
`makedist`Create a probability distribution object using specified parameter values.
`fitdist`Fit a probability distribution object to sample data.
Distribution FitterFit a probability distribution to sample data using the interactive Distribution Fitter app and export the fitted object to the workspace.
`paretotails`Create a piecewise distribution object that has generalized Pareto distributions in the tails.

## Output Arguments

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cdf values, returned as a scalar value or an array of scalar values. `y` is the same size as `x` after any necessary scalar expansion. Each element in `y` is the cdf value of the distribution, specified by the corresponding elements in the distribution parameters (`A`, `B`, `C`, and `D`) or the probability distribution object (`pd`), evaluated at the corresponding element in `x`.

## Alternative Functionality

• `cdf` is a generic function that accepts either a distribution by its name `'name'` or a probability distribution object `pd`. It is faster to use a distribution-specific function, such as `normcdf` for the normal distribution and `binocdf` for the binomial distribution. For a list of distribution-specific functions, see Supported Distributions.

• 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.