m

2

'

distribution quantifies the likelihood that its value lies

in any given interval. Two commonly used distri-

j

2

butions, the normal and the lognormal, are described

'

later in this appendix.

The summation form above involving the *X*i term

random variables in the Taylor's series or point

provides the variance of a population containing

estimate methods, one must provide values of their

exactly *N *elements. Usually, a *sample *of size *N *is

expected values and standard deviations, which are

used to obtain an *estimate of the variance *of the

two of several probabilistic *moments *of a random

associated random variable which represents an *entire*

variable. These can be calculated from data or esti-

mated from experience. For random variables which

obtain an unbiased estimate of the population working

are not independent of each other, but tend to vary

from a finite sample, the *N *is replaced by *N *- 1:

together, correlation coefficients must also be

assigned.

j

2

(1) Mean value. The *mean *value X of a set of *N*

measured values for the random variable *X *is obtained

by summing the values and dividing by *N*:

(4) Standard deviation. To express the scatter or

dispersion of a random variable about its expected

j *X*i

value in the same units as the random variable itself,

the *standard deviation *FX is taken as the square root of

X '

the variance; thus:

Fx ' *Var X*

(2) Expected value. The *expected value E[X] *of a

random variable is the mean value one would obtain if

all possible values of the random variable were multi-

(5) Coefficient of variation. To provide a

plied by their likelihood of occurrence and summed.

convenient dimensionless expression of the uncertainty

Where a mean value can be calculated from

inherent in a random variable, the standard deviation is

representative data, it provides an unbiased estimate of

divided by the expected value to obtain the *coefficient*

the expected value of a parameter; hence, the mean

and expected value are numerically the same. The ex-

pected value is defined as:

FX

100 %

m

The expected value, standard deviation, and coefficient

of variation are interdependent: knowing any two, the

where

third is known. In practice, a convenient way to

estimate moments for parameters where little data are

available is to assume that the coefficient of variation is

continuous random variables)

similar to previously measured values from other data

sets for the same parameter.

random variables)

(3) Variance. The *variance Var [X] *of a random

be correlated or independent; if correlated, the like-

variable *X *is the expected value of the squared

lihood of a certain value of the random variable *Y*

difference between the random variable and its mean

depends on the value of the random variable *X*. For

value. Where actual data are available, the variance of

example, the strength of sand may be correlated with

the data can be calculated by subtracting each value

density or the top blanket permeability may be corre-

from the mean, squaring the result, and determining

lated with grain size of the sand. The *covariance*

the average of these values:

B-4

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