# Distributions, Statistical Operators and Histogram Functions

This chapter

This chapter provides a more elaborate description of the distributions and distribution and sample operators listed in this table, this table and this table. You can use this information when you want to set up an experiment around your (optimization-based) AIMMS model.

Description of distributions

For each of the available distributions we describe

• its parameters, mean and variance,

• the unit relationship between its parameters and result,

• its shape, and

• its typical use in applications.

Such information may be useful in the selection and use of a distribution to describe the particular statistical behavior of input data of experiments that you want to perform on top of your model. However, a general guideline for choosing the right might be in order and is provided in the next paragraph.

Choosing the right distribution

Whenever your experiment counts a number of occurrences, you should first make a distinction between experiments with replacement (i.e. throwing dice), experiments without replacement (i.e. drawing cards from a deck), or experiments in which independent occurrences take place at random moments (i.e. customers appearing at a desk). Having made this distinction, Overview of discrete distributions in AIMMS will help you to select the right distribution for your experiment. In any other case the `Normal` distribution should be considered first. Although this distribution is unbounded, it is declining so rapidly that it can often be used even when the result should be bounded. If the `Normal` distribution does not suffice, the primary selection criterium is existence of bounds: AIMMS provides the user with distributions with no bounds, one (lower) bound and two (upper and lower) bounds. See Continuous Distributions (continuous distributions) for details.

Description of distribution operators

For each of the available distribution and sample operators we provide

• the interpretation of its result, and

• the formula for the computation of the operator.

Such information may be useful when you want to perform an analysis of the results of your experiments.

Option for backward compatibility

All distribution operators that are listed in Distribution Operators have been introduced in AIMMS 3.4, although the `DistributionCumulative` and `DistributionInverseCumulative` operator were already available under the names `CumulativeDistribution` and `InverseCumulativeDistribution`, respectively. Furthermore, in order to obtain a consistent set of distribution functions the prototype for some of them has been slightly adapted. Continuous Distributions discusses the function prototype of the continuous distribution functions in full detail. Both the old and the new function prototypes are discussed in the AIMMS Function Reference. To make sure that models using distribution functions and developed in an older version of AIMMS are working correctly, you should set the option `Distribution_compatibility` to ‘`AIMMS 3.0`’.