# Random Variables

In order to carry out a Probabilistic Analysis with *Slide3*, you must define at least ONE (or more) of your model input parameters, as RANDOM VARIABLES. This is done using the options in the Statistics menu. In *Slide3*, material properties can be defined as Random Variables.

## What is a Random Variable?

A Random Variable in *Slide3*, is any model input parameter that you have selected and defined a statistical distribution for, using the options in the Statistics menu in the project settings. For example, a Random Variable could be:

- A material property such as Cohesion, Friction Angle, Unit Weight, or the Shear Strength of the material

## Defining Random Variables

Any number or combination of input parameters, may be defined as Random Variables, for a *Slide3* Probabilistic Analysis.

After you have selected an input parameter to be a Random Variable, you must enter the following parameters, to define the probability density function (PDF) for the Random Variable:

- Statistical Distribution (i.e. Normal, Uniform, Triangular, etc)
- Standard deviation (if applicable for the type of Statistical Distribution)
- Minimum and Maximum values
- Mean Value

Note if defining shear strength as random variable, then you need only enter either 1) Statistical Distribution and COV (coefficient of variation), or; 2) Statistical Distribution and Minimum and Maximum values. This depends on the distribution you have selected.

For detailed information about selecting and defining Random Variables, for each of the different types of model input parameters in Slide, see the following topics:

The general features common to all Random Variables in *Slide3* are discussed below.

## Statistical Distribution

A Statistical Distribution must be chosen for each Random Variable in Slide. The type of Statistical Distribution, together with the mean, standard deviation and minimum / maximum values, determines the shape and extent of the probability density function you are defining for the Random Variable.

There are several different Statistical Distributions available in *Slide3*, for defining Random Variables. In most cases, a Normal or Lognormal distribution will distribution will be used. However, several other distribution types are available. These range from simple Uniform or Triangular distributions, to more complex distributions such as Beta and Gamma, which allow the user to model virtually any type of statistical distribution likely to be encountered in geotechnical engineering.

For further information, see the Statistical Distributions Overview topic.

## Standard Deviation

The Standard Deviation of a Random Variable, is a measure of the variance or scatter of the variable about the Mean value. The larger the Standard Deviation, then the wider the range of values which the Random Variable may assume (within the limits of the Minimum and Maximum values). NOTE:

- The Standard Deviation is applicable for Normal, Lognormal, Beta and Gamma distributions.
- It is NOT APPLICABLE for Uniform, Triangular, or Exponential distributions. If you are using one of these distributions, then you will NOT be able to enter a Standard Deviation.
- For tips on estimating values of Standard Deviation, see the Normal Distribution topic.
- In the case of the shear strength random variable, coefficient of variation (COV) is entered instead of Standard Deviation. COV is defined as the Standard Deviation divided by the Mean value, which is 1.0 in the shear strength case.

## Minimum / Maximum Values

For each Random Variable, you must define a Minimum and Maximum allowable value. It is important to note that, for the purposes of data input, the Minimum / Maximum values are specified as RELATIVE quantities (i.e. as distances from the Mean), rather than as absolute values. This simplifies the data input for the user, and is much less prone to error.

During the analysis, the Relative Minimum and Maximum values are converted to the actual Minimum and Maximum values, when the statistical sampling is carried out for each Random Variable, as follows:

MINIMUM = MEAN â€“ Relative MINIMUM

MAXIMUM = MEAN + Relative MAXIMUM

EXAMPLE: if the Mean Friction Angle = 35, and the Relative Minimum = Relative Maximum = 10, then the actual Minimum = 25 degrees, and the actual Maximum = 45 degrees.

- For each Random Variable, you must always specify non-zero values for the Relative Minimum and the Relative Maximum. If BOTH the Relative Minimum and Relative Maximum are equal to zero, no statistical samples will be generated for that variable, and the value of the variable will always be equal to the Mean.
- In most cases, if you are using a Normal distribution (or other distribution which is symmetric about the Mean), the Relative Minimum and Relative Maximum values will be equal. However, they do not necessarily have to be equal, if your distribution is not symmetric.
- For the Water Table location or the Tension Crack location, the Minimum and Maximum locations are specified graphically, by drawing boundaries on the model.
- The Minimum and Maximum values are applicable for ALL Statistical Distributions in Slide3.
- IMPORTANT NOTE: In the case of the shear strength random variable,
*absolute*min and max are used for the applicable distributions: Uniform, Triangular, Exponential.

## Mean

The Mean represents the average value of the Random Variable. Note that the Mean value of a Random Variable is equal to the Deterministic value of the variable that has been entered, in the main Slide input data dialogs.

- For example, if you select the Cohesion of a given Material, to be a Random Variable in the Material Statistics dialog, then the Mean value of Cohesion is automatically equal to the Cohesion of the Material which has been specified in the Define Material Properties dialog (in the Properties menu).

You may change the Mean (or Deterministic) value of a variable, using either the options in the Statistics menu, or the main Slide data input dialogs. Just remember, if you change the value in one dialog (e.g. the Material Statistics dialog), it will be changed in the other (e.g. the Define Material Properties dialog), and vice versa.

In the case of the shear strength random variable, the random variable is a shear strength factor that is applied to the calculated shear strength. As such the mean value is automatically 1.0 and does not need to be defined by the user.