the combined effect of how two variables vary

the normal or lognormal distribution which has the

together. The definition of the covariance is:

expected value and standard deviation assumed. This

can easily be done with spreadsheet software.

(2) Figure B-1 illustrated probability density

functions for the random variables *c *and N. A

which is equivalent to:

probability density function has the property that for

any *X*, the value of *f(x) *is proportional to the like-

lihood of *X*. The area under a probability density

mm

function is unity. The probability that the random

variable *X *lies between two values *X*1 and *X*2 is the

integral of the probability density function taken

In the above equation, *f(X,Y) *is the joint probability

between the two values. Hence:

density function of the random variables *X *and *Y*. To

calculate the covariance from data, the following

equation can be used:

m

j *X*i & X Yi & Y

1

(3) The *cumulative distribution function CDF *or

To provide a nondimensional measure of the degree of

correlation between *X *and *Y*, the *correlation coefficient*

function from minus infinity to *X*:

DX,Y is obtained by dividing the covariance by the

product of the standard deviations:

m

DX,*Y *'

FX FY

&4

The correlation coefficient may assume values from

-1.0 to +1.0. A value of 1.0 or -1.0 indicates there is

random variable *X *is less than the given *x*.

perfect linear correlation; given a value of *X*, the value

of *Y *is known and hence is not random. A value of

zero indicates no linear correlation between variables.

suggested method to assign or check assumed

A positive value indicates the variables increase and

moments for random variables is to:

decrease together; a negative value indicates that one

variable decreases as the other increases. Pairs of

Assume trial values for the expected value and

standard deviation and take the random

coefficients.

variable to be normal or lognormal.

Plot the resulting density function and tabulate

and plot the resulting cumulative distribution

function (spreadsheet software is a convenient

way to do this).

(1) The terms *probability distribution *and *proba-*

Assess the reasonableness of the shape of the

a function that defines a continuous random variable.

The Taylor's series and point estimate methods

Repeat above steps with successively

described herein to determine moments of performance

improved estimates of the expected value and

functions require only the mean and standard deviation

standard deviation until an appropriate *pdf *and

of random variables and their correlation coefficients;

knowledge of the form of the probability density

function is not necessary. However, in order to ensure

that estimates made for these moments are reasonable,

distribution is the most well-known and widely

it is recommended that the engineer plot the shape of

B-5

Integrated Publishing, Inc. |