After the simulations were run, the samples were sorted according to PPD and represented as scatterplots alongside the

probability distribution of the PPD.

The rth moment of a discrete

probability distribution about X = 0 is defined by

In probability theory, a log-normal distribution is a continuous

probability distribution of a random variable whose logarithm is normally distributed.

The

probability distribution of [N.sub.PM] for selected scenarios (FFS, PSR) and initial scenarios (short for OPTI) are shown in Fig.

Caption: Figure 4:

Probability distribution of normal force chains.

Apart from the joint

probability distribution, copulas are also able to establish conditional

probability distributions.

In order to display the space

probability distribution, we will transform (15) from original spherical coordinates to the popular Cartesian coordinates through the coordinate transformations r = [square root of ([x.sup.2] + [y.sup.2] + [z.sup.2])] and cos [theta] = z/r and then obtain the corresponding space

probability distribution [rho](x, y, z).

According to the statistics, it is found that the

probability distribution of displacement efficiency follows the normal distribution N([mu], [[sigma].sup.2]), where [mu] is the mean of the normal distribution and [sigma] is the standard deviation of the normal distribution.

Then, the joint

probability distribution is given by applying probability multiplication and additive rules:

First, in terms of definitions, we follow the ecological literature, defining diversity as the interplay of "richness" and "evenness" in a

probability distribution. Richness refers to the number of different diversity types in a system.

The Monte Carlo method simulates the full system many times, each time randomly choosing a value for each variable from its

probability distribution [15].

A

probability distribution is a statistical model that shows the possible outcomes of a particular event or course of action as well as the statistical likelihood of each event.