cdf

Cumulative distribution function

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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.

example

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.

example

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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Compute Normal Distribution cdf by Specifying Distribution Name and Parameters

Open Live Script

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 $μ$ equal to 1 and the standard deviation $σ$ 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

Open Live Script

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 $μ$ equal to 1 and the standard deviation $σ$ 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

Open Live Script

Create a Poisson distribution object with the rate parameter, $λ$ , 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, $λ$ .

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

Open Live Script

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)

Figure contains an axes object. The axes object contains an object of type line.

Plot Gamma Distribution cdf

Open Live Script

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;

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent a = 1, b = 1, a = 1, b = 2, a = 2, b = 1.

Fit Pareto Tails to t Distribution and Compute the cdf

Open Live Script

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)

Compute the cdf at the values in q.

cdf(obj,q)

ans = 2×1

    0.1000
    0.9000

Input Arguments

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`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]