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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 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

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

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```

References

[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.