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Operations Research Models and Methods
Computation Section
Subunit Discrete Distributions
 - Geometric

The geometric distribution relates to a series of independent Bernoulli trials. The random variable is the number of failures before the first success is observed. The single parameter p is the probability of success on any one trial.

Example: If you don’t win or lose on the first roll of the craps game, you might wonder how long the game will last. Assume the dice totals to 4 on the first roll. The player will continue to roll the dice until a 4 shows, and the player wins, or a 7 shows, and the player loses. Based on the probabilities computed earlier for a roll of a pair of dice, on any given roll, the game will end with probability 0.25 and continue with probability 0.75.

We define the random variable as the number of rolls prior to the last roll. That number may be 0, 1, 2,… which is random variable described by the geometric distribution. The results on the left show that the mean number of rolls is 3 and the variance is 12. The first six probabilities of the distribution are computed at the left. The game might continue for quite a few rolls. The probability that more than 5 rolls is required is about 0.18.



An alternative statement of the random variable for the geometric distribution is the number of trials until the first success. The count includes the success as well as the previous failures. This random variable must be at least 1, and the distribution for this statement is shown at the left.
The figure shows the example modeled with the alternative statement. The distribution is shifted by 1. The mean is increased by 1, but the variance and the other moments remain the same.
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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved

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