Documentation

# BirnbaumSaundersDistribution

Birnbaum-Saunders probability distribution object

## Description

A `BirnbaumSaundersDistribution` object consists of parameters, a model description, and sample data for a Birnbaum-Saunders probability distribution.

The Birnbaum-Saunders distribution was originally proposed as a lifetime model for materials subject to cyclic patterns of stress and strain, where the ultimate failure of the material comes from the growth of a prominent flaw. In materials science, Miner's Rule suggests that the damage occurring after n cycles, at a stress level with an expected lifetime of N cycles, is proportional to n / N. Whenever Miner's Rule applies, the Birnbaum-Saunders model is a reasonable choice for a lifetime distribution model.

The Birnbaum-Saunders distribution uses the following parameters.

ParameterDescriptionSupport
`beta`scale parameterβ > 0
`gamma`shape parameterγ > 0

## Creation

There are several ways to create a `BirnbaumSaundersDistribution` probability distribution object.

## Properties

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### Distribution Parameters

Scale parameter of the Birnbaum-Saunders distribution, specified as a positive scalar value.

Data Types: `single` | `double`

Shape parameter of the Birnbaum-Saunders distribution, specified as a positive scalar value.

Data Types: `single` | `double`

### Distribution Characteristics

Logical flag for truncated distribution, specified as a logical value. If `IsTruncated` equals `0`, the distribution is not truncated. If `IsTruncated` equals `1`, the distribution is truncated.

Data Types: `logical`

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: `double`

Covariance matrix of the parameter estimates, specified as a p-by-p matrix, where p is the number of parameters in the distribution. The (`i`,`j`) element is the covariance between the estimates of the `i`th parameter and the `j`th parameter. The (`i`,`i`) element is the estimated variance of the `i`th parameter. If parameter `i` is fixed rather than estimated by fitting the distribution to data, then the (`i`,`i`) elements of the covariance matrix are 0.

Data Types: `double`

Logical flag for fixed parameters, specified as an array of logical values. If `0`, the corresponding parameter in the `ParameterNames` array is not fixed. If `1`, the corresponding parameter in the `ParameterNames` array is fixed.

Data Types: `logical`

Distribution parameter values, specified as a vector.

Data Types: `single` | `double`

Truncation interval for the probability distribution, specified as a vector containing the lower and upper truncation boundaries.

Data Types: `single` | `double`

### Other Object Properties

Probability distribution name, specified as a character vector.

Data Types: `char`

Data used for distribution fitting, specified as a structure containing the following:

• `data`: Data vector used for distribution fitting.

• `cens`: Censoring vector, or empty if none.

• `freq`: Frequency vector, or empty if none.

Data Types: `struct`

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: `char`

Distribution parameter names, specified as a cell array of character vectors.

Data Types: `char`

## Object Functions

 `cdf` Cumulative distribution function `icdf` Inverse cumulative distribution function `iqr` Interquartile range `mean` Mean of probability distribution `median` Median of probability distribution `negloglik` Negative loglikelihood of probability distribution `paramci` Confidence intervals for probability distribution parameters `pdf` Probability density function `proflik` Profile likelihood function for probability distribution `random` Random numbers `std` Standard deviation of probability distribution `truncate` Truncate probability distribution object `var` Variance of probability distribution

## Examples

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Create a Birnbaum-Saunders distribution object using the default parameter values.

`pd = makedist('BirnbaumSaunders')`
```pd = BirnbaumSaundersDistribution Birnbaum-Saunders distribution beta = 1 gamma = 1 ```

Create a Birnbaum-Saunders distribution object by specifying the parameter values.

`pd = makedist('BirnbaumSaunders','beta',2,'gamma',5)`
```pd = BirnbaumSaundersDistribution Birnbaum-Saunders distribution beta = 2 gamma = 5 ```

Compute the mean of the distribution.

`m = mean(pd)`
```m = 27 ```