# Random Variables

In order to carry out a Probabilistic Analysis with Slide, 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. Almost all model input parameters in Slide, can be defined as Random Variables, for example, material properties, support properties, load magnitudes, water table location etc.

What is a Random Variable ?

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

• A material property such as Cohesion, Friction Angle or Unit Weight

• A support property such as Tensile Capacity

• Water Table location

Defining Random Variables

Any number or combination of input parameters, may be defined as Random Variables, for a Slide Probabilistic Analysis. For example, you might define certain material properties to be Random Variables, as well as the Water Table location. Or you might define a Load Magnitude as a Random Variable.

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

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:

Material Statistics

Support Statistics

Water Table Statistics

Tension Crack Statistics

Piezo Line Statistics

The general features common to all Random Variables in Slide 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 Slide, 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.

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 – 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 Slide.

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.