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Operations Research Models and Methods
Computation Section
Random Variables

Discrete Distributions

  A discrete probability distribution function is completely described by the set of possible values the random variable can take and by the probabilities assigned to each value. On this page we describe the general features of discrete distributions. On the following pages we describe a variety of named distributions. All are available from the Random Variables add-in. We use the triangular distribution pictured below for an example.


The mathematical notation for a discrete distribution is shown at the left. The values associated with a distribution are often integer, but in general they need not be. For a discrete distribution, only the values in the set X have nonzero probabilities and these must be nonnegative. For a valid distribution, summing the probabilities over the set X must yield the value 1.

The number of values in X may be finite or infinite. When the number is infinite, the set must be countable infinite. An example is the set of all nonnegative integers.

When the possible values are integers, we will often use k rather than x as the notation for the values. We use PDF to refer to the Probability Distribution Function.

  The example experiment involves throwing a pair of standard dice. Each die has the numbers {1,2,3,4,5,6}, so the sum of the two dice ranges from 2 through 12. The value with the greatest probability is called the mode, so 7 is the mode of this distribution. The probabilities sum to 1 and all probabilities are nonnegative, so this is a valid distribution.
The Random Variables add-in defines distributions using named ranges on the worksheet. For the example, the range B2:B5 has the name Dice. The Mean and Variance in B6 and B7, as well as the probabilities in B9 through B19, are computed with user-defined functions provided by the add-in.




Several quantities can be computed from the PDF that describe simple characteristics of the distribution. These are called moments. The most common is the mean, the first moment about the origin, and the variance, the second moment about the mean. The mean is a measure of the centrality of the distribution and the variance is a measure of the spread of the distribution about the mean.

The skewness is computed from the third moment about the mean. This quantity can be positive or negative. We normalize the measure by squaring the third moment and dividing it by the third power of the variance. To recover the sign of the third moment, we multiply this ratio by the sign of the third moment. The skewness indicates whether the distribution has a long tail to the right of the mean (positive) or to the left (negative). The skewness is 0 for a symmetric distribution.

The kurtosis is a measure of the thickness of the tails of the distribution. The use of this measure is not obvious in most cases, but it is included for completeness. The formula for this measure subtracts 3 from the ratio of the fourth moment about the mean and the square of the variance. The Normal distribution has a kurtosis of 3, so this normalization provides a value relative to the value for the Normal distribution. It can be positive (greater than the Normal) or negative (less than the Normal).

The moments for the dice example are computed with user-defined functions functions provided by the add-in.


We use distributions to answer questions about situations that involve random variables. We use the game of Craps to illustrate the use of the triangular distribution. In this game, the player roles a pair of dice. We assume the player is female. If on the first roll of the dice the player throws a 7 or 11, she wins. If the player throws 2, 3 or 12, she loses. If the player throws a number other than 2, 3, 7, 11, or 12, the number thrown is called the point. If the player does not win or lose on the first roll, she must roll the dice again and continue to roll until she throws the point and wins, or a 7, and loses. The triangular distribution describes a single roll of the dice. Since the alternatives are mutually exclusive, probabilities of an event involving several different results are obtained by summing. We compute the probabilities associated with the first throw at the left. We use the Craps game as an example for several other distributions on the following pages.


Named Distributions

  It is useful for modeling purposes to know about the named discrete distributions. When an experiment on which a random variable is based satisfies the logical conditions associated with a named distribution, the probability values for the random variable are immediately determined. Then we can use the distribution without extensive experimentation to answer decision questions about the situation. We consider a number of named distributions on the following pages. Click on a link at the far left for descriptions and examples.

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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved

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