# pdf

Probability density functions

## Syntax

• y = pdf('name',x,A) example
• y = pdf('name',x,A,B) example
• y = pdf('name',x,A,B,C)

## Description

example

y = pdf('name',x,A) returns the probability density function (pdf) for the one-parameter distribution family specified by 'name', evaluated at the values in x. A contains the parameter value for the distribution.

example

y = pdf('name',x,A,B) returns the pdf for the two-parameter distribution family specified by 'name', evaluated at the values in x. A and B contain the parameter values for the distribution.

y = pdf('name',x,A,B,C) returns the pdf for the three-parameter distribution family specified by 'name', evaluated at the values in x. A, B, and C contain the parameter values for the distribution.

example

y = pdf(pd,x) returns the probability density function of the probability distribution object, pd, evaluated at the values in x.

## Examples

collapse all

### Compute the Normal Distribution pdf

Create a standard normal distribution object with the mean equal to 0 and the standard deviation equal to 1.

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

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

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

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

y = pdf(pd,x)
y =

0.0540    0.2420    0.3989    0.2420    0.0540

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 1, the corresponding pdf value y is equal to 0.2420.

Alternatively, you can compute the same pdf values without creating a probability distribution object. Use the pdf function, and specify a standard normal distribution using the same parameter values for and .

y2 = pdf('Normal',x,mu,sigma)
y2 =

0.0540    0.2420    0.3989    0.2420    0.0540

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

### Compute the Poisson Distribution pdf

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

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

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

x = [0 1 2 3 4];

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

y = pdf(pd,x)
y =

0.1353    0.2707    0.2707    0.1804    0.0902

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 3, the corresponding pdf value in y is equal to 0.1804.

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

y2 = pdf('Poisson',x,lambda)
y2 =

0.1353    0.2707    0.2707    0.1804    0.0902

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

## Input Arguments

collapse all

### 'name' — Probability distribution nameprobability distribution name string

Probability distribution name, specified as one of the following probability distribution name strings.

nameDistributionInput Parameter AInput Parameter BInput Parameter C
'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
'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: log meanσ: log scale parameter
'Lognormal'Lognormal Distributionμ: log meanσ: log standard deviation
'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
'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

### x — Values at which to evaluate pdfscalar value | array of scalar values

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

• If x is a scalar value, and if you specify distribution parameters A, B, or C as arrays, then cdf expands x into a constant matrix the same size as A and B.

• If x is an array, and if you specify distribution parameters A, B, or C as arrays, then x, A, B, and C must all be the same size.

Example: [0.1,0.25,0.5,0.75,0.9]

Data Types: single | double

### A — First probability distribution parameterscalar value | array of scalar values

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

If x and A are arrays, they must be the same size. If x is a scalar, then cdf expands it into a constant matrix the same size as A. If A is a scalar, then cdf expands it into a constant matrix the same size as x.

Data Types: single | double

### B — Second probability distribution parameterscalar value | array of scalar values

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

If x, A, and B are arrays, they must be the same size. If x is a scalar, then cdf expands it into a constant matrix the same size as A and B. If A or B are scalars, then cdf expands them into constant matrices the same size as x

Data Types: single | double

### C — Third probability distribution parameterscalar value | array of scalar values

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

If x, A, B, and C are arrays, they must be the same size. If x is a scalar, then cdf expands it into a constant matrix the same size as A, B, and C. If any of A, B or C are scalars, then cdf expands them into constant matrices the same size as x.

Data Types: single | double

### pd — Probability distributionprobability distribution object

Probability distribution, specified as a probability distribution object created using one of the following.

 makedist Create a probability distribution object using specified parameter values. fitdist Fit a probability distribution object to sample data. dfittool Fit a probability distribution object to sample data using the interactive Distribution Fitting app. paretotails Create a Pareto tails object. gmdistribution Create a Gaussian mixture distribution object.

## Output Arguments

collapse all

### y — Probability density functionarray

Probability density function of the specified probability distribution, returned as an array.

• If you specify distribution parameters A, B, or C, then y is the common size of x, A, B, and C after any necessary scalar expansion.

• If you specify a probability distribution object, pd, then y has the same dimensions as x.