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Lognormal mean and variance

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

`[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 parameters in
`sigma`

must be positive.

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

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

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

[1] Mood, A. M., F. A. Graybill, and D. C.
Boes. *Introduction to the Theory of Statistics.* 3rd
ed., New York: McGraw-Hill, 1974. pp. 540–541.