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Operations Research Models and Methods
Computation Section
Subunit Random Variables
 - Linear Transform

Named distributions are defined over specific ranges of the random variable. For examples, the binomial distribution is defined on the nonnegative integers, the exponential distribution is defined over the nonegative real numbers and the Beta distribution is defined on the range [0, 1]. In applications, it is often useful to make linear transformations of random variables that are controllable though parameters on the worksheet. This shifts the mean, changes the variance and may change the sign of the skewness. Formulas for probabilities and inverse probabilities are also adjusted. When the Linear Transform button on the distribution dialog is checked, two additional parameters, a and b are included on the form. The add-in user-defined functions use the transformation parameters to adjust the moments, probabilities and inverse probabilities as shown to the left.

The example below shows three distributions all based on the same Beta distribution. The first case makes no transformation because a is 0 and b is 1. The second case multiplies the random variable by 2 and adds 2 to the result. The distribution is shifted and spread out. The mean and variance change, but the skewness and kurtosis remain the same. The third case multiplies the random variable by -1. This flips the distribution to the range [-1,0]. The mean changes in sign, the variance remains the same and the skewness changes sign. The kurtosis is not affected by a transformation. The ranges of probability statements and the values of inverse probabilities are changed by the transformation.

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

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