cdf
Cumulative distribution function
Syntax
Description
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 equal to 1 and the standard deviation 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 equal to 1 and the standard deviation 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, , 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, .
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 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 |
'Pearson' | Pearson Distribution | μ mean | σ standard deviation | γ skewness | κ kurtosis |
'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 |
'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 |
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 |
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 namename
or a probability distribution objectpd
. It is faster to use a distribution-specific function, such asnormcdf
for the normal distribution andbinocdf
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, includecoder.Constant('Normal')
in the-args
value ofcodegen
(MATLAB Coder).The input argument
pd
can be a fitted probability distribution object for beta, exponential, extreme value, lognormal, normal, and Weibull distributions. Createpd
by fitting a probability distribution to sample data from thefitdist
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 R2006aR2023b: Support for Pearson distributions
Starting in R2023b, cdf
supports Pearson distributions.
See Also
pdf
| ecdf
| icdf
| mle
| random
| makedist
| fitdist
| Distribution Fitter | paretotails
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