### Introduction

As defined previously, a large part of the modelling is defined using probability distributions (observation model, inter-individual variability, …). An explicit definition of a random variable can be provided in the block DEFINITION:.

### Distribution definition

A random variable can be defined through its probability distribution. It is a list including a distribution name with specific parameters. Obviously, distribution is a keyword. The elements of this list are identified according to their names (the order does not matter).

`Mlxtran`

only supports distributions based on the normal distribution. More precisely, we consider that there exists a monotonic transformation such that is normally distributed:

- normal distribution, used with keyword normal, and
- log-normal distribution, used with keyword lognormal, and
- logit-normal distribution, used with keyword logitnormal, and
- probit-normal distribution, used with keyword probitnormal, and , where is the cumulative distribution function of the distribution.

These distributions are defined by the mean and standard deviation of the normal distribution, i.e of the transformed variable . Therefore, when we say that is log-normally distributed with a mean and a standard variation , it means that

Here, and are the mean and standard deviation of . Notice that mean and sd are keywords defining the the mean and standard deviation of the normal distribution.

### Example

To define the previous example, X log-normally distributed with a mean and a standard variation , in `Mlxtran`

as a individual parameter, one just has to write

input= {mu, sigma} DEFINITION: X = {distribution=lognormal, mean=mu, sd=sigma}

Then, from a syntax point of view :

In that case, the mean definition in the distribution is the mean of the transformed value, i.e. in the previous example. Therefore, we introduce a keyword typical which corresponds to the “typical” value of X around which you want to make the transformation. From a syntax point of view, we have

Therefore, the use of the keywords typical and mean depends if you are in the transformed referential or not.