       Functions - Moments  The enumeration method computes the moments of the function by enumerating all possible values of the random variables and using probability theory to compute the moments. For a discrete distribution with integer intervals, the evaluation is exact. The formulas for the mean and variance of a function of integer-random variables are below. To compute the mean and variance all possible values of the variables are generated and the resultant sums are compiled. We use the second form of the variance computation in the add-in. On selecting on the Moments command from the menu, the add-in presents the following dialog. The first field accepts the name of a previously constructed function form. During the enumeration, the Excel calculation mode is controlled by the program. The Calculation frame of the dialog determines when the workbook is calculated. For the Automatic option, the workbook recalculates every time a cell is changed. This is the default for most Excel users, but it may result in slow computations when there are several random variables and the number of combinations is very large. For the Worksheet and Workbook options, the add-in sets the calculation mode to manual. Then after the complete variable vector is constructed, the worksheet or workbook is calculated depending on which option is chosen.

For the example case, the program computes the total number in the set X and when that number is greater than 1000 presents the number with an opportunity to continue or not. After a few seconds, the enumeration process is complete and the results are placed on the form. The mean and variance are accurate in the example case, because the entire probability space has been enumerated and there is no approximation of the function values. The 1 in cell B13 indicates this. The computed moments are in rows 17 and 18. Other Distributions

The first example used the same distribution for all random variables. Distributions are easily changed by changing the distribution names and parameters as illustrated below. The F_2 form was constructed in the manner of the first example, but the parameters of the Binomial for the first two random variables were changed on the worksheet. The third random variable was changed to a Bernoulli distribution by inserting "Bern" in cell L3. The single parameter for the Bernoulli is placed in L4. Similarly, the fourth random variable was changed to a Normal distribution by placing "Norm" in M3. The mean and standard deviation parameters for the Normal are in M4 and M5, respectively. For most distributions, the first three letters of the name are sufficient to identify the distribution and the parameters are placed immediately below the distribution names. Of course, the words placed in column I to identify the parameters are no longer appropriate for the new distributions. Since the moments are computed with a discrete enumeration process, functional values for continuous distributions must be approximated.

The add-in computes probabilities accurately, but the function value for the interval is taken as the value at the midpoint of the interval. The accuracy of the approximation depends on the size of the interval (0.5 for the example Normal distribution). Of course the results of the moments analysis are also approximate.

Since the range of the Normal distribution is infinite and the example considers only the range from 0 to 5, the enumeration does not cover the entire probability. Rather we see 0.99404 in cell J13. The formulas below show how the mean and variance are adjusted by the total probability. The adjustment assumes probabilities of 0 outside the range. Using the features described on this page almost any function of random variables can be analyzed to find its mean and variance. When the variables are discrete, the enumeration results are accurate. When some of the distributions are continuous, the results are approximate.

The number of computations required by the enumeration process depends on the number of random variables, the range for each and the intervals into which the range is divided. The total number computations can be very large, perhaps too large for reasonable computation times. An alternative to the enumeration approach is the Monte Carlo simulation method discussed on the next page.  Operations Research Models and Methods
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by Paul A. Jensen    