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

Consider an experiment that involves n independent Bernoulli trials. The random variable is the sum of the of the n Bernoulli random variables. This is called the Binomial random variable. The variable has n+1 possible integer values ranging from 0 to n. Its PDF is the binomial distribution which has two parameters, p and n.


Example: : The reliability of a computer is defined as the probability of successful operation throughout a particular mission. A study determines that the reliability for a given mission as 0.9. Because the mission is very important and computer failure is extremely serious, we provide five identical computers for this mission. The computers operate independently and the failure or success of one does not affect the probability of failure or success of the others. Our job is to compute the probability of mission success, or system reliability, under the following three operating rules:

a. All five computers must work for mission success
b. At least three out of five must work for mission success
c. At least one computer must work for mission success


The success or failure of each computer is a Bernoulli random variable with 1 representing success and 0 representing failure. The probability of success is p, and we assume that the computers are independent with respect to failure. The number of working computers, x, is the random variable of interest, and the binomial distribution, with parameters n = 5 and p = 0.9, is the appropriate PDF.

The solution of the problem is computed above using functions provided by the add-in. These results show the value of redundancy for increasing reliability. In case a, none of the computers is redundant since all are required for successful operation. In case b, we say that two computers are redundant since only three are required. In case c, all but one are redundant. The reliability of the system increases as redundancy is increased

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

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