Random Variables

To carry out a Probabilistic Analysis with RocFall3, 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, the Define Material Properties menu, and the Define Seeder Properties menu. In RocFall3, material properties and seeder properties can be defined as Random Variables.

What is a Random Variable?

A Random Variable in RocFall3 is any model input parameter that you have selected and defined with a statistical distribution. For example, a Random Variable could be:

• A material property such as the normal coefficient of restitution or the friction angle
• A seeder property such as the rock mass or the initial velocity

Defining Random Variables

Any number or combination of input parameters may be defined as Random Variables.

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

Statistical Distribution

A Statistical Distribution must be chosen for each Random Variable. 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 RocFall3 for defining Random Variables. In most cases, a Normal or Lognormal distribution would 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, the wider the range of values which the Random Variable may assume (within the limits of the Minimum and Maximum values).

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.
• The Minimum and Maximum values are applicable for ALL Statistical Distributions in RocFall3.

Mean

The Mean represents the average value of the Random Variable.

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