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Lognormal Distribution

Overview

The lognormal distribution is a probability distribution whose logarithm has a normal distribution. It is sometimes called the Galton distribution. The lognormal distribution is applicable when the quantity of interest must be positive, since log(x) exists only when x is positive.

Parameters

The lognormal distribution uses these parameters.

ParameterDescriptionSupport
muMean of logarithmic values<μ<
sigmaStandard deviation of logarithmic valuesσ0

To fit the lognormal distribution to data and find the parameter estimates, use lognfit, fitdist, or mle.

  • For uncensored data, lognfit and fitdist find the unbiased estimates of the distribution parameters, and mle finds the maximum likelihood estimates.

  • For censored data, lognfit, fitdist, and mle find the maximum likelihood estimates.

Unlike lognfit and mle, which return parameter estimates, fitdist returns the fitted probability distribution object LognormalDistribution. The object properties mu and sigma store the parameter estimates.

Probability Density Function

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

Descriptive Statistics

The mean is

The variance is

You can compute these descriptive statistics using the lognstat function.

Relationship to Other Distributions

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.

Examples

Compute the Lognormal Distribution pdf

Suppose the income of a family of four in the United States follows a lognormal distribution with mu = log(20,000) and sigma = 1. Compute and plot the income density.

x = (10:1000:125010)';
y = lognpdf(x,log(20000),1.0);

figure;
plot(x,y)
h = gca;
h.XTick = [0 30000 60000 90000 120000];
h.XTickLabel = {'0','$30,000','$60,000',...
                    '$90,000','$120,000'};

Consulte también

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