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# lognpdf

Lognormal probability density function

## Syntax

`Y = lognpdf(X,mu,sigma)`

## Description

`Y = lognpdf(X,mu,sigma)` returns values at `X` of the lognormal pdf with distribution parameters `mu` and `sigma`. `mu` and `sigma` are the mean and standard deviation, respectively, of the associated normal distribution. `X`, `mu`, and `sigma` can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of `Y`. A scalar input for `X`, `mu`, or `sigma` is expanded to a constant array with the same dimensions as the other inputs.

The lognormal pdf is

`$y=f\left(x|\mu ,\sigma \right)=\frac{1}{x\sigma \sqrt{2\pi }}{e}^{\frac{-{\left(\mathrm{ln}x-\mu \right)}^{2}}{2{\sigma }^{2}}}$`

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:

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

So, a lognormal distribution with mean m and variance v has parameters

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

If you do not know the population mean and variance, m and v, for the lognormal distribution, you can estimate $\mu$ and $\sigma$ in the following way:

```mu = mean(log(X)) sigma = std(log(X))```

The lognormal distribution is applicable when the quantity of interest must be positive, since log(X) exists only when X is positive.

## Examples

collapse all

Compute the pdf of a lognormal distribution with `mu = 0` and `sigma = 1`.

```x = (0:0.02:10); y = lognpdf(x,0,1); ```

Plot the pdf.

```plot(x,y); grid; xlabel('x'); ylabel('p') ```

## 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.