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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
`mu`Mean of logarithmic values$-\infty <\mu <\infty$
`sigma`Standard deviation of logarithmic values$\sigma \ge 0$

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'};```