# `stats`::`lognormalPDF`

Probability density function of the log-normal distribution

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## Syntax

```stats::lognormalPDF(`m`, `v`)
```

## Description

`stats::lognormalPDF(m, v)` returns a procedure representing the probability density function of the lognormal distribution with location parameter `m` and shape parameter `v`.

A random variable X is log-normally distributed if ln(X) is a normally distributed variable. The “location parameter” m of X is the mean of ln(X) and the “shape parameter” v is the variance of ln(X).

The procedure `f := stats::lognormalPDF(m, v)` can be called in the form `f(x)` with an arithmetical expression `x`. The value is returned.

If `x` is a floating-point number and both `m` and `v` can be converted to floating-point numbers, this value is returned as a floating-point number. Otherwise, a symbolic expression is returned.

Numerical values for m and v are only accepted if they are real and v is positive.

## Environment Interactions

The function is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

We compute the probability density with location parameter m = 2 and shape parameter v = 4 at various points:

```f := stats::lognormalPDF(2, 4): f(-infinity), f(-3), f(2.0), f(PI), f(infinity)```
` `
`delete f:`

### Example 2

We use symbolic arguments:

```f := stats::lognormalPDF(m, v): f(x), f(0.4)```
` `

When numerical values are assigned to m and v, the function f starts to produce numerical values:

```m := PI: v := 2: f(3), f(3.0)```
` `
`delete f, m, v:`

### Example 3

The following plot shows the influence of the shape parameter on the log-normal distribution:

```plotfunc2d(stats::lognormalPDF(1, 0.25)(x), stats::lognormalPDF(1, 0.5)(x), stats::lognormalPDF(1, 1)(x), stats::lognormalPDF(1, 2)(x), stats::lognormalPDF(1, 4)(x), stats::lognormalPDF(1, 8)(x), x = -0.5 .. 4, ViewingBoxYRange = 0 .. 1.1, LegendVisible = FALSE)``` Due to its logarithmic influence, the location parameter changes the shape of the distribution, too:

```plotfunc2d(stats::lognormalPDF(m, 0.5)(x) \$ m = 0.5..2 step 0.5, x = -0.5 ..4, ViewingBoxYRange = 0 .. 0.5, LegendVisible = FALSE)``` ## Parameters

 `m` The location parameter: an arithmetical expression representing a real value `v` The shape parameter: an arithmetical expression representing a positive real value

## Return Values

#### Mathematical Modeling with Symbolic Math Toolbox

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