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

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.

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.

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

Compute Normal Distribution cdf by Specifying Distribution Name and Parameters

Compute the cdf values for a normal distribution by specifying the distribution name 'Normal' and the distribution parameters.

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 normal distribution with the mean $$\mu$$ equal to 1 and the standard deviation $$\sigma$$ equal to 5.

mu = 1;
sigma = 5;
y = cdf('Normal',x,mu,sigma)

y = 1×5

    0.2743    0.3446    0.4207    0.5000    0.5793

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 0.5000.

Compute Normal Distribution cdf Using Distribution Object

Create a normal distribution object and compute the cdf values of the normal distribution using the object.

Create a normal distribution object with the mean $$\mu$$ equal to 1 and the standard deviation $$\sigma$$ equal to 5.

mu = 1;
sigma = 5;
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 normal distribution at the values in x.

y = cdf(pd,x)

y = 1×5

    0.2743    0.3446    0.4207    0.5000    0.5793

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 0.5000.

Compute Poisson Distribution cdf

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.

Plot Standard Normal Distribution cdf

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)

Plot Gamma Distribution cdf

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 t Distribution and Compute the cdf

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

name — Probability distribution name

character vector or string scalar of probability distribution name

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

name	Distribution	Input Parameter A	Input Parameter B	Input Parameter C	Input Parameter D
'Beta'	Beta Distribution	a first shape parameter	b second shape parameter	N/A	N/A
'Binomial'	Binomial Distribution	n number of trials	p probability of success for each trial	N/A	N/A
'BirnbaumSaunders'	Birnbaum-Saunders Distribution	β scale parameter	γ shape parameter	N/A	N/A
'Burr'	Burr Type XII Distribution	α scale parameter	c first shape parameter	k second shape parameter	N/A
'Chisquare' or 'chi2'	Chi-Square Distribution	ν degrees of freedom	N/A	N/A	N/A
'Exponential'	Exponential Distribution	μ mean	N/A	N/A	N/A
'Extreme Value' or 'ev'	Extreme Value Distribution	μ location parameter	σ scale parameter	N/A	N/A
'F'	F Distribution	ν1 numerator degrees of freedom	ν2 denominator degrees of freedom	N/A	N/A
'Gamma'	Gamma Distribution	a shape parameter	b scale parameter	N/A	N/A
'Generalized Extreme Value' or 'gev'	Generalized Extreme Value Distribution	k shape parameter	σ scale parameter	μ location parameter	N/A
'Generalized Pareto' or 'gp'	Generalized Pareto Distribution	k tail index (shape) parameter	σ scale parameter	μ threshold (location) parameter	N/A
'Geometric'	Geometric Distribution	p probability parameter	N/A	N/A	N/A
'Half Normal' or 'hn'	Half-Normal Distribution	μ location parameter	σ scale parameter	N/A	N/A
'Hypergeometric' or 'hyge'	Hypergeometric Distribution	m size of the population	k number of items with the desired characteristic in the population	n number of samples drawn	N/A
'InverseGaussian'	Inverse Gaussian Distribution	μ scale parameter	λ shape parameter	N/A	N/A
'Logistic'	Logistic Distribution	μ mean	σ scale parameter	N/A	N/A
'LogLogistic'	Loglogistic Distribution	μ mean of logarithmic values	σ scale parameter of logarithmic values	N/A	N/A
'LogNormal'	Lognormal Distribution	μ mean of logarithmic values	σ standard deviation of logarithmic values	N/A	N/A
'Loguniform'	Loguniform Distribution	a lower endpoint (minimum)	b upper endpoint (maximum)	N/A	N/A
'Nakagami'	Nakagami Distribution	μ shape parameter	ω scale parameter	N/A	N/A
'Negative Binomial' or 'nbin'	Negative Binomial Distribution	r number of successes	p probability of success in a single trial	N/A	N/A
'Noncentral F' or 'ncf'	Noncentral F Distribution	ν1 numerator degrees of freedom	ν2 denominator degrees of freedom	δ noncentrality parameter	N/A
'Noncentral t' or 'nct'	Noncentral t Distribution	ν degrees of freedom	δ noncentrality parameter	N/A	N/A
'Noncentral Chi-square' or 'ncx2'	Noncentral Chi-Square Distribution	ν degrees of freedom	δ noncentrality parameter	N/A	N/A
'Normal'	Normal Distribution	μ mean	σ standard deviation	N/A	N/A
'Pearson'	Pearson Distribution	μ mean	σ standard deviation	γ skewness	κ kurtosis
'Poisson'	Poisson Distribution	λ mean	N/A	N/A	N/A
'Rayleigh'	Rayleigh Distribution	b scale parameter	N/A	N/A	N/A
'Rician'	Rician Distribution	s noncentrality parameter	σ scale parameter	N/A	N/A
'Stable'	Stable Distribution	α first shape parameter	β second shape parameter	γ scale parameter	δ location parameter
'T'	Student's t Distribution	ν degrees of freedom	N/A	N/A	N/A
'tLocationScale'	t Location-Scale Distribution	μ location parameter	σ scale parameter	ν shape parameter	N/A
'Uniform'	Uniform Distribution (Continuous)	a lower endpoint (minimum)	b upper endpoint (maximum)	N/A	N/A
'Discrete Uniform' or 'unid'	Uniform Distribution (Discrete)	n maximum observable value	N/A	N/A	N/A
'Weibull' or 'wbl'	Weibull Distribution	a scale parameter	b shape parameter	N/A	N/A

Example: 'Normal'

x — Values at which to evaluate cdf

scalar value | array of scalar values

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

A — First probability distribution parameter

scalar value | array of scalar values

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

B — Second probability distribution parameter

scalar value | array of scalar values

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

C — Third probability distribution parameter

scalar value | array of scalar values

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

D — Fourth probability distribution parameter

scalar value | array of scalar values

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

pd — Probability distribution

probability distribution object

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

Distribution Object	Function or App to Create Probability Distribution Object
BetaDistribution	makedist, fitdist, Distribution Fitter
BinomialDistribution	makedist, fitdist, Distribution Fitter
BirnbaumSaundersDistribution	makedist, fitdist, Distribution Fitter
BurrDistribution	makedist, fitdist, Distribution Fitter
EmpiricalDistribution	fitdist
ExponentialDistribution	makedist, fitdist, Distribution Fitter
ExtremeValueDistribution	makedist, fitdist, Distribution Fitter
GammaDistribution	makedist, fitdist, Distribution Fitter
GeneralizedExtremeValueDistribution	makedist, fitdist, Distribution Fitter
GeneralizedParetoDistribution	makedist, fitdist, Distribution Fitter
HalfNormalDistribution	makedist, fitdist, Distribution Fitter
InverseGaussianDistribution	makedist, fitdist, Distribution Fitter
KernelDistribution	fitdist, Distribution Fitter
LogisticDistribution	makedist, fitdist, Distribution Fitter
LoglogisticDistribution	makedist, fitdist, Distribution Fitter
LognormalDistribution	makedist, fitdist, Distribution Fitter
LoguniformDistribution	makedist
MultinomialDistribution	makedist
NakagamiDistribution	makedist, fitdist, Distribution Fitter
NegativeBinomialDistribution	makedist, fitdist, Distribution Fitter
NormalDistribution	makedist, fitdist, Distribution Fitter
PearsonDistribution	makedist
Piecewise distribution with generalized Pareto distributions in the tails	paretotails
PiecewiseLinearDistribution	makedist
PoissonDistribution	makedist, fitdist, Distribution Fitter
RayleighDistribution	makedist, fitdist, Distribution Fitter
RicianDistribution	makedist, fitdist, Distribution Fitter
StableDistribution	makedist, fitdist, Distribution Fitter
tLocationScaleDistribution	makedist, fitdist, Distribution Fitter
TriangularDistribution	makedist
UniformDistribution	makedist
WeibullDistribution	makedist, fitdist, Distribution Fitter

Output Arguments

y — cdf values

scalar value | array of scalar values

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The input argument name must be a compile-time constant. For example, to use the normal distribution, include coder.Constant('Normal') in the -args value of codegen (MATLAB Coder).

• The input argument pd can be a fitted probability distribution object for beta, exponential, extreme value, lognormal, normal, and Weibull distributions. Create pd by fitting a probability distribution to sample data from the fitdist function. For an example, see Code Generation for Probability Distribution Objects.

For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2023b: Support for Pearson distributions

Starting in R2023b, cdf supports Pearson distributions.