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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 the following parameters.

ParameterDescriptionSupport
muLog mean
sigmaLog standard deviation

### 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)
set(gca,'xtick',[0 30000 60000 90000 120000])
set(gca,'xticklabel',{'0','\$30,000','\$60,000',...
'\$90,000','\$120,000'})
```

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