icdf

Inverse cumulative distribution function

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Syntax

x = icdf(name,p,A)

x = icdf(name,p,A,B)

x = icdf(name,p,A,B,C)

x = icdf(name,p,A,B,C,D)

x = icdf(pd,p)

Description

x = icdf(name,p,A) returns the inverse cumulative distribution function (icdf) for the one-parameter distribution family specified by name and the distribution parameter A, evaluated at the probability values in p.

x = icdf(name,p,A,B) returns the icdf for the two-parameter distribution family specified by name and the distribution parameters A and B, evaluated at the probability values in p.

example

x = icdf(name,p,A,B,C) returns the icdf for the three-parameter distribution family specified by name and the distribution parameters A, B, and C, evaluated at the probability values in p.

x = icdf(name,p,A,B,C,D) returns the icdf for the four-parameter distribution family specified by name and the distribution parameters A, B, C, and D, evaluated at the probability values in p.

x = icdf(pd,p) returns the icdf function of the probability distribution object pd, evaluated at the probability values in p.

example

Examples

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Compute Normal Distribution icdf by Specifying Distribution Name and Parameters

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

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

p = [0.1,0.25,0.5,0.75,0.9];

Compute the icdf 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 = icdf('Normal',p,mu,sigma)

y = 1×5

   -5.4078   -2.3724    1.0000    4.3724    7.4078

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 1, the corresponding icdf value y is equal to 7.4078.

Compute Normal Distribution icdf Using Distribution Object

Open Live Script

Create a normal distribution object and compute the icdf 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 p to contain the probability values at which to calculate the icdf.

p = [0.1,0.25,0.5,0.75,0.9];

Compute the icdf values for the normal distribution at the values in p.

x = icdf(pd,p)

x = 1×5

   -5.4078   -2.3724    1.0000    4.3724    7.4078

Each value in x corresponds to a value in the input vector p. For example, at the value p equal to 0.9, the corresponding icdf value x is equal to 7.4078.

Compute the Poisson Distribution icdf

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 p to contain the probability values at which to calculate the icdf.

p = [0.1,0.25,0.5,0.75,0.9];

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

x = icdf(pd,p)

x = 1×5

     0     1     2     3     4

Each value in x corresponds to a value in the input vector p. For example, at the value p 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 $λ$ .

x2 = icdf('Poisson',p,lambda)

x2 = 1×5

     0     1     2     3     4

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

Compute Standard Normal Critical Values

Open Live Script

Create a standard normal distribution object.

pd = makedist('Normal')

pd = 
  NormalDistribution

  Normal distribution
       mu = 0
    sigma = 1

Determine the critical values at the 5% significance level for a test statistic with a standard normal distribution, by computing the upper and lower 2.5% values.

x = icdf(pd,[.025,.975])

x = 1×2

   -1.9600    1.9600

Plot the cdf and shade the critical regions.

p = normspec(x,0,1,'outside')

Figure contains an axes object. The axes object with title Probability Outside Limits is 0.05, xlabel Critical Value, ylabel Density contains an object of type line. This object represents Normal.

p = 
0.0500

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
`'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'

`p` — Probability values at which to evaluate icdf
scalar value | array of scalar values

Probability values at which to evaluate the icdf, specified as a scalar value, or an array of scalar values in the range [0,1].

If one or more of the input arguments p, A, B, C, and D are arrays, then the array sizes must be the same. In this case, icdf 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]