The game of Craps is a popular gambling game. In this game, the player rolls a pair of dice and sums the numbers showing. A sum of 7 or 11 wins for the player, and a sum of 2, 3, or 12 loses. Any other number is called the point. The player then rolls the dice again. If she rolls the point number, she wins. If she throws a 7, she loses. Any other number requires another roll. The game continues until the gambler rolls a 7 or her point number. We use this situation as an example of a random variable.

 Simple probably analysis determines that the sum of two standard dice is a discrete random variable with integer values ranging from 2 through 12. The probability distribution of the sum is the triangular distribution with a mode of 7. To define this distribution for subsequent computation, select the Add_RV item from the OR_MM menu. The dialog box shown below opens.

Sixteen named distributions are available with one user defined distribution and a general distribution. The option buttons at the left select discrete distributions, while the option buttons on the right select continuous distributions. An exception is General that can define both continuous and discrete distributions. The User distribution allows entry of a user specified discrete probability distribution.

For the example we name the random variable Dice, identify it as an integer triangular distribution, specify its lower value, mode and upper value as 2, 7 and 12 respectively. The check boxes on the right of the dialog, list optional information that will be shown with the distribution.

One or more random variables must be defined with the Add_RV option before any of the other programs or functions may be used. The names for random variables must be unique and not previously defined for the workbook. The cell reference shown at the top of the dialog box is the upper left corner of the worksheet range where the random variable information will be stored. The value of this cell is the reference to the cursor location when the dialog is called. When the "Show Titles" checkbox is clicked, the titles associated with the distribution parameters are shown. When not clicked only the parameters are shown without titles. The latter is useful when a series of random variables are defined each with the same set of parameters. The other checkboxes at the right specify that a series of computations related to the random variable are to be presented along with the definition.

 After identifying a distribution and defining appropriate parameters, the random variable information is placed on the worksheet with the rectangular construction as in the example. For each random variable defined in this manner, an Excel Name is assigned to that portion of the array that identifies the distribution type and parameters. For the example the four cells headed by "Triang_Int" are given the name "Dice". The name must obey Excel restrictions on naming regions of the worksheet. The random variable is referenced by this name. Any number of random variables may be included on the worksheet. Once defined, the distribution parameters may be changed by simply typing a new value. Distribution parameters may also be functions determined by other cells on the worksheet. The example also shows additional information regarding the random variable. These are obtained with user defined functions, to be described later. The orientation of the distribution data is determined by the Orient frame of the dialog. This example has the vertical orientation.

The figure below shows the same distribution with a horizontal orientation.

Replications

It is often convenient to create several random variables simultaneously. To do this select the number of replications to a number greater than 1. The figure below shows three Normally distributed random variables in the vertical orientation. The three distributions are differentiated by the _1, _2 or _3 added to their names.

The horizontal orientation for three replications is illustrated below.

User Distribution

To create a User Distribution, place the cursor on the sheet where you want the distribution to appear. Choose Add RV from the OR_MM menu, enter the name and number of cells defining the distribution.

The resulting display with 5 cells is shown on the left. Once the table is defined, the user enters the several possible values of the random variable and the associated values of the probability distribution as in the example. The distribution values should sum to 1, however, the functions will normalize the distribution if the total probability is not 1. The table on the right shows three user distributions defined by setting the number of replications to 3. User distributions can only have vertical orientations.

General Distribution

It is often convenient to construct a form first and add the distribution data later. The General Distribution form is for this purpose. To define the distribution click the General button. The parameter values can be defined here, but it is convenient to leave them at 0.

An empty form is created on the worksheet as below.

The columns can be used to identify any of the named distributions listed on the distribution dialog. In the following, Binomial, Beta and Exponential distributions are defined.

It is unnecessary to enter the entire title of the distribution because in all but two cases only the first three letters are required. For example "Binomial" could be entered as "Bin". The two cases are the uniform-integer and triangular-it distributions. These are identified by "uni-int" and "tri-int", respectively. Other designations are possible as long as the first three letters are "uni" or "tri" and "int" appears in the name.

The titles in the left column do not indicate the names of the parameters because each distribution has different parameters. The parameters are entered in the cells immediately following the distribution title in the order specified on the distribution dialog. For example, the four parameters for the Beta distribution are alpha, beta, lower bound and upper bound.

Operations Research Models and Methods
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by Paul A. Jensen