# A Petri Net Approach for Business Process Modeling and Simulation
## Appendix A Probability Distributions  
Probability distributions definition. _-Describes how the results are expected to vary. Because these distributions are based on the expectation of something happening, they turn out to be useful models for making inferences and making decisions under conditions of uncertainty._  
_-A mathematical function that describes how the possible values of a random variable are distributed over a set of possible outcomes; in other words, it shows the probabilities associated with different values that a random variable can take on in a given experiment or phenomenon [39]._  
**Types of probability distributions**[39]. Several different types of probability distributions are mentioned below. **Normal Distribution (Gaussian). ** Known as the Gaussian bell, it is widely used due to the central limit theorem. This distribution is used to model data that are continuous and symmetric about the mean. For a sample of random numbers to follow a normal distribution, it must have the following characteristics:  
* Symmetry: in a normal distribution, the probability density curve is symmetric about its mean. This means that the values above the mean are equally symmetric to the values below the mean. * Mean, median, and mode coincide: in a normal distribution, the mean, median, and mode are equal and lie at the center of the distribution. This contributes to the symmetry of the distribution. * Bell shape: the shape of a normal distribution resembles a smooth and continuous bell. **Uniform Distribution. ** This distribution is used to model equally likely events in a continuous range; all values within the range have the same probability of occurring. For a sample of random numbers to follow a uniform distribution, it must have the following characteristics:  
* Equal probability: all values in the range have the same probability of occurring. This means that there is no bias towards any specific value. * Defined range: the uniform distribution is defined in a specific range. All possible values within this range have the same probability. * Mean value rule: in a uniform distribution, the mean (average) value of the distribution is equal to the sum of the endpoints of the interval divided by 2, that is, if a and b are the endpoints of the interval, then the mean value is (a + b)/2. * Linear transformations: when taking a random variable that follows a uniform distribution and applying a linear transformation (for example, multiplication or addition), the result is a uniform distribution in the new range. **Triangular Distribution. ** This distribution models random variables with values that are restricted to a specified interval, and is approximately triangular in shape. This distribution is narrowly skewed in situations where there is a lack of data. For a sample of random numbers to follow a triangular distribution, it must have the following characteristics:  
* Triangular shape: the triangular distribution is characterized by having a triangular shape in its probability density graph, meaning that the probability density is at its maximum at a point within an interval and decreases linearly towards the ends of the interval. * Defined minimum, maximum, and mode value: in a triangular distribution, three main parameters must be defined: the minimum value (a), the maximum value (b), and the mode (c). These parameters determine the shape and range of the distribution. * Optional symmetry: a triangular distribution can be symmetric or asymmetric depending on the location of mode (c) in relation to the interval (min, max). If c is in the center of the interval, the distribution is symmetric, while if c is closer to one of the extremes, the distribution is skewed. **Gamma Distribution. ** This is a generalization of the exponential distribution, and is used to model more general wait times.