# lognrnd

Lognormal random numbers

## Syntax

``r = lognrnd(mu,sigma)``
``r = lognrnd(mu,sigma,sz1,...,szN)``
``r = lognrnd(mu,sigma,sz)``

## Description

example

````r = lognrnd(mu,sigma)` generates a random number from the lognormal distribution with the distribution parameters `mu` (mean of logarithmic values) and `sigma` (standard deviation of logarithmic values).```
````r = lognrnd(mu,sigma,sz1,...,szN)` generates an array of lognormal random numbers, where `sz1,...,szN` indicates the size of each dimension.```

example

````r = lognrnd(mu,sigma,sz)` generates an array of lognormal random numbers, where vector `sz` specifies `size(r)`.```

## Examples

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Find the distribution parameters from the mean and variance of a lognormal distribution and generate a lognormal random value from the distribution.

Find the distribution parameters `mu` and `sigma` from the mean and variance.

```m = 1; % mean v = 2; % variance mu = log((m^2)/sqrt(v+m^2))```
```mu = -0.5493 ```
`sigma = sqrt(log(v/(m^2)+1))`
```sigma = 1.0481 ```

Generate a lognormal random value.

```rng('default') % For reproducibility r = lognrnd(mu,sigma)```
```r = 1.0144 ```

Save the current state of the random number generator. Then create a 1-by-5 vector of lognormal random numbers from the lognormal distribution with the parameters 3 and 10.

```s = rng; r = lognrnd(3,10,[1,5])```
```r = 1×5 109 × 0.0000 1.8507 0.0000 0.0001 0.0000 ```

Restore the state of the random number generator to `s`, and then create a new 1-by-5 vector of random numbers. The values are the same as before.

```rng(s); r1 = lognrnd(3,10,[1,5])```
```r1 = 1×5 109 × 0.0000 1.8507 0.0000 0.0001 0.0000 ```

Create a matrix of lognormally distributed random numbers with the same size as an existing array.

```A = [3 2; -2 1]; sz = size(A); R = lognrnd(0,1,sz)```
```R = 2×2 1.7120 0.1045 6.2582 2.3683 ```

You can combine the previous two lines of code into a single line.

`R = lognrnd(1,0,size(A));`

## Input Arguments

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Mean of logarithmic values for the lognormal distribution, specified as a scalar value or an array of scalar values.

To generate random numbers from multiple distributions, specify `mu` and `sigma` using arrays. If both `mu` and `sigma` are arrays, then the array sizes must be the same. If either `mu` or `sigma` is a scalar, then `lognrnd` expands the scalar argument into a constant array of the same size as the other argument. Each element in `r` is the random number generated from the distribution specified by the corresponding elements in `mu` and `sigma`.

Example: `[0 1 2; 0 1 2]`

Data Types: `single` | `double`

Standard deviation of logarithmic values for the lognormal distribution, specified as a nonnegative scalar value or an array of nonnegative scalar values.

If `sigma` is zero, then the output `r` is always equal to `exp(mu)`.

To generate random numbers from multiple distributions, specify `mu` and `sigma` using arrays. If both `mu` and `sigma` are arrays, then the array sizes must be the same. If either `mu` or `sigma` is a scalar, then `lognrnd` expands the scalar argument into a constant array of the same size as the other argument. Each element in `r` is the random number generated from the distribution specified by the corresponding elements in `mu` and `sigma`.

Example: `[1 1 1; 2 2 2]`

Data Types: `single` | `double`

Size of each dimension, specified as separate arguments of integers. For example, specifying `5,3,2` generates a 5-by-3-by-2 array of random numbers from the lognormal probability distribution.

If either `mu` or `sigma` is an array, then the specified dimensions `sz1,...,szN` must match the common dimensions of `mu` and `sigma` after any necessary scalar expansion. The default values of `sz1,...,szN` are the common dimensions.

• If you specify a single value `sz1`, then `r` is a square matrix of size `sz1`-by-`sz1`.

• If the size of any dimension is `0` or negative, then `r` is an empty array.

• Beyond the second dimension, `lognrnd` ignores trailing dimensions with a size of 1. For example, `lognrnd``(mu,sigma,3,1,1,1)` produces a 3-by-1 vector of random numbers.

Example: `5,3,2`

Data Types: `single` | `double`

Size of each dimension, specified as a row vector of integers. For example, specifying `[5 3 2]` generates a 5-by-3-by-2 array of random numbers from the lognormal probability distribution.

If either `mu` or `sigma` is an array, then the specified dimensions `sz` must match the common dimensions of `mu` and `sigma` after any necessary scalar expansion. The default values of `sz` are the common dimensions.

• If you specify a single value `[sz1]`, then `r` is a square matrix of size `sz1`-by-`sz1`.

• If the size of any dimension is `0` or negative, then `r` is an empty array.

• Beyond the second dimension, `lognrnd` ignores trailing dimensions with a size of 1. For example, `lognrnd``(mu,sigma,[3,1,1,1])` produces a 3-by-1 vector of random numbers.

Example: `[5 3 2]`

Data Types: `single` | `double`

## Output Arguments

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Lognormal random numbers, returned as a scalar value or an array of scalar values with the dimensions specified by `sz1,...,szN` or `sz`. Each element in `r` is the random number generated from the distribution specified by the corresponding elements in `mu` and `sigma`.

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### Lognormal Distribution

The lognormal distribution is a probability distribution whose logarithm has a normal distribution.

The mean m and variance v of a lognormal random variable are functions of the lognormal distribution parameters µ and σ:

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

Also, you can compute the lognormal distribution parameters µ and σ from the mean m and variance v:

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

## Alternative Functionality

• `lognrnd` is a function specific to lognormal distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `random`, which supports various probability distributions. To use `random`, create a `LognormalDistribution` probability distribution object and pass the object as an input argument or specify the probability distribution name and its parameters. Note that the distribution-specific function `lognrnd` is faster than the generic function `random`.

• To generate random numbers interactively, use `randtool`, a user interface for random number generation.

 Marsaglia, G., and W. W. Tsang. “A Fast, Easily Implemented Method for Sampling from Decreasing or Symmetric Unimodal Density Functions.” SIAM Journal on Scientific and Statistical Computing. Vol. 5, Number 2, 1984, pp. 349–359.

 Evans, M., N. Hastings, and B. Peacock. Statistical Distributions. 2nd ed., Hoboken, NJ: John Wiley & Sons, Inc., 1993.