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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 Fitting 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.

This property is read-only.

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
icdfInverse cumulative distribution function of probability distribution object
iqrInterquartile range of probability distribution object
median Median of probability distribution object
pdfProbability density function of probability distribution object
randomGenerate random numbers from probability distribution object
truncateTruncate probability distribution object
meanMean of probability distribution object
negloglikNegative log likelihood of probability distribution object
paramciConfidence intervals for probability distribution parameters
proflikProfile likelihood function for probability distribution object
std Standard deviation of probability distribution object
varVariance of probability distribution object

Definitions

Lognormal Distribution

The lognormal distribution is closely related to the normal distribution. If x is distributed lognormally with parameters μ and σ, then log(x) is distributed normally with mean μ and standard deviation σ. The lognormal distribution is applicable when the quantity of interest must be positive, since log(x) exists only when x is positive.

The lognormal distribution uses the following parameters.

ParameterDescriptionSupport
muLog mean<μ<
sigmaLog standard deviationσ0

The probability density function (pdf) of the lognormal distribution is

f(x|μ,σ)=1xσ2πexp{(lnxμ)22σ2};x>0.

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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