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# stats::lognormalCDF

Cumulative distribution function of the log-normal distribution

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

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

## Description

stats::lognormalCDF(m, v) returns a procedure representing the cumulative distribution function

of the log-normal 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::lognormalCDF(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 evaluate the CDF of a lognormal distribution for some arbitrary parameter values:

```f := stats::lognormalCDF(1/2, 3/4):
f(0.1), f(10.3)```

`delete f:`

### Example 2

We use symbolic arguments:

```f := stats::lognormalCDF(m, v):
f(3), f(x)```

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

```m := 4: v := PI:
f(3), f(3.0)```

`delete f, m, v:`

### Example 3

From the definition of "lognormal" above it is clear that the probability of X < 0 is zero for X lognormally distributed:

`plotfunc2d(stats::lognormalCDF(0,1))`

The following plot shows the influence of the shape parameter on the shape of the lognormal distribution:

```f03 := stats::lognormalCDF(0, 0.3):
f1  := stats::lognormalCDF(0, 1):
f3  := stats::lognormalCDF(0, 3):
f9  := stats::lognormalCDF(0, 9):
plotfunc2d(f03, f1, f3, f9,  x = 0..10)```

As for the normal distribution, a larger value of the shape parameter stretches the lognormal distribution, also changing its shape in the process:

```plotfunc2d(stats::lognormalCDF(m, 1)\$ m = 0..2 step .1,
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