# Documentation

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# prob.LognormalDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

Lognormal probability distribution object

## Description

prob.LognormalDistribution is an object consisting of parameters, a model description, and sample data for a lognormal probability distribution.

Create a probability distribution object with specified parameter values using makedist. Alternatively, fit a distribution to data using fitdist or the Distribution Fitter app.

## Construction

pd = makedist('Lognormal') creates a lognormal probability distribution object using the default parameter values.

pd = makedist('Lognormal','mu',mu,'sigma',sigma) creates a lognormal probability distribution object using the specified parameter values.

### Input Arguments

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Log mean for the lognormal distribution, specified as a scalar value. mu is the mean of the log of x, when x has a lognormal distribution.

Data Types: single | double

Log standard deviation for the lognormal distribution, specified as a nonnegative scalar value. sigma is the standard deviation of the log of x, when x has a lognormal distribution.

Data Types: single | double

## Properties

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Log mean for the lognormal distribution, stored as a scalar value.

Data Types: single | double

Log standard deviation for the lognormal distribution, stored as a nonnegative scalar value.

Data Types: single | double

Probability distribution name, stored as a character vector. This property is read-only.

Data Types: char

Data used for distribution fitting, stored 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

Logical flag for truncated distribution, stored as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated. This property is read-only.

Data Types: logical

Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only.

Data Types: single | double

Covariance matrix of the parameter estimates, stored 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 ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith 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. This property is read-only.

Data Types: single | double

Distribution parameter descriptions, stored as a cell array of character vectors. Each cell contains a short description of one distribution parameter. This property is read-only.

Data Types: char

Logical flag for fixed parameters, stored 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. This property is read-only.

Data Types: logical

Distribution parameter names, stored as a cell array of character vectors. This property is read-only.

Data Types: char

Distribution parameter values, stored as a vector. This property is read-only.

Data Types: single | double

Truncation interval for the probability distribution, stored as a vector containing the lower and upper truncation boundaries. This property is read-only.

Data Types: single | double

## Methods

### Inherited Methods

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

## Examples

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Create a lognormal distribution object using the default parameter values.

pd = makedist('Lognormal')
pd =

LognormalDistribution

Lognormal distribution
mu = 0
sigma = 1

Create a lognormal distribution object by specifying the parameter values.

pd = makedist('Lognormal','mu',5,'sigma',2)
pd =

LognormalDistribution

Lognormal distribution
mu = 5
sigma = 2

Compute the mean of the lognormal distribution.

mean(pd)
ans =

1.0966e+03

The mean of the lognormal distribution is not equal to the mu parameter.

Generate random numbers from the lognormal distribution and compute their log values.

rng(47);  % for reproducibility
x = random(pd,10000,1);
logx = log(x);

Compute the mean of the log values.

m = mean(logx)
m =

5.0156

The mean of the log of x is equal to the mu parameter of x, since x has a lognormal distribution.

Plot logx.

histogram(logx,50)

The plot shows that the log values of x are normally distributed with a mean equal to 5 and a standard deviation equal to 2.

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