The PEM of Rosenbleuth (1975) is described here in detail for a function of two random

variables *y*(*x*1,*x*2). The correlation coefficient of random variables *x*1 and *x*2 is defined as

ρ=

(A1)

where *S*x is the standard deviation of *x *and *Cov*(*x*1,*x*2) is the covariance. A correlation

coefficient of 1 indicates a perfect linear correlation between the variables, and a coeffi-

cient of 0 indicates perfectly uncorrelated variables. Numerical experiments have shown

that if the relationship between random variables over a limited range can be written as

(A2)

then the magnitude of ρ will be approximately 1.

To simplify this discussion, we will assume that the distribution of each random vari-

able is symmetric about the mean. The point estimates that represent the distribution of

each random variable are then

1+ ρ

(A3)

4

1 ρ

4

These point estimates are weighting factors that sum to 1 and, in the case of uncorrelated

random variables, are each 1/4. The function *y*(*x*1,*x*2) is evaluated at points that are a

standard deviation from the mean of each random variable :

(A4)

Then, the expected values or moments of the function can be obtained as

(A5)

The expected value or mean of *y *is found by setting *n *= 1. The variance of *y *is readily

obtained as

2

2

(A6)

The minimum and maximum values of *y *fall outside the limits of the values obtained in eq

A4. Rosenbleuth (1975) discussed the generalization of this method to functions of any

number of random variables and random variables with asymmetric distributions.

In summary, the PEM provides estimates of the mean, variance, and limits of the

distribution of a function of random variables. These four parameters uniquely specify a

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