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# lognstat

Lognormal mean and variance

## Syntax

[M,V] = lognstat(mu,sigma)

## Description

[M,V] = lognstat(mu,sigma) returns the mean of and variance of the lognormal distribution with parameters mu and sigma. mu and sigma are the mean and standard deviation, respectively, of the associated normal distribution. mu and sigma can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of M and V. A scalar input for mu or sigma is expanded to a constant array with the same dimensions as the other input.

The normal and lognormal distributions are closely related. If X is distributed lognormally with parameters µ and σ, then log(X) is distributed normally with mean µ and standard deviation σ.

The mean m and variance v of a lognormal random variable are functions of µ and σ that can be calculated with the lognstat function. They are:

$\begin{array}{l}m=\mathrm{exp}\left(\mu +{\sigma }^{2}/2\right)\\ v=\mathrm{exp}\left(2\mu +{\sigma }^{2}\right)\left(\mathrm{exp}\left({\sigma }^{2}\right)-1\right)\end{array}$

A lognormal distribution with mean m and variance v has parameters

$\begin{array}{l}\mu =\mathrm{log}\left({m}^{2}/\sqrt{v+{m}^{2}}\right)\\ \sigma =\sqrt{\mathrm{log}\left(v/{m}^{2}+1\right)}\end{array}$

## Examples

Generate one million lognormally distributed random numbers with mean 1 and variance 2:

```m = 1;
v = 2;
mu = log((m^2)/sqrt(v+m^2));
sigma = sqrt(log(v/(m^2)+1));

[M,V]= lognstat(mu,sigma)
M =
1
V =
2.0000

X = lognrnd(mu,sigma,1,1e6);

MX = mean(X)
MX =
0.9974
VX = var(X)
VX =
1.9776```