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98 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

8. Use the table for the cumulative distribution function of a standard normal distribution to calcu-
late probabilities.
9. Approximate probabilities for some binomial and Poisson distributions.
CD MATERIAL
10. Use continuity corrections to improve the normal approximation to those binomial and Poisson
distributions.

Answers for most odd numbered exercises are at the end of the book. Answers to exercises whose
numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete
worked solutions to certain exercises are also available in the e-Text. These are indicated in the
Answers to Selected Exercises section by a box around the exercise number. Exercises are also
available for some of the text sections that appear on CD only. These exercises may be found within
the e-Text immediately following the section they accompany.

4-1 CONTINUOUS RANDOM VARIABLES

Previously, we discussed the measurement of the current in a thin copper wire. We noted that
the results might differ slightly in day-to-day replications because of small variations in vari-
ables that are not controlled in our experiment—changes in ambient temperatures, small im-
purities in the chemical composition of the wire, current source drifts, and so forth.
Another example is the selection of one part from a day’s production and very accurately
measuring a dimensional length. In practice, there can be small variations in the actual
measured lengths due to many causes, such as vibrations, temperature fluctuations, operator
differences, calibrations, cutting tool wear, bearing wear, and raw material changes. Even the
measurement procedure can produce variations in the final results.
In these types of experiments, the measurement of interest—current in a copper wire ex-
periment, length of a machined part—can be represented by a random variable. It is reason-
able to model the range of possible values of the random variable by an interval (finite or
infinite) of real numbers. For example, for the length of a machined part, our model enables
the measurement from the experiment to result in any value within an interval of real numbers.
Because the range is any value in an interval, the model provides for any precision in length
measurements. However, because the number of possible values of the random variable X is
uncountably infinite, X has a distinctly different distribution from the discrete random vari-
ables studied previously. The range of X includes all values in an interval of real numbers; that
is, the range of X can be thought of as a continuum.
A number of continuous distributions frequently arise in applications. These distributions
are described, and example computations of probabilities, means, and variances are provided
in the remaining sections of this chapter.

4-2 PROBABILITY DISTRIBUTIONS AND PROBABILITY
DENSITY FUNCTIONS

Density functions are commonly used in engineering to describe physical systems. For exam-
ple, consider the density of a loading on a long, thin beam as shown in Fig. 4-1. For any point
x along the beam, the density can be described by a function (in grams/cm). Intervals with
large loadings correspond to large values for the function. The total loading between points a
and b is determined as the integral of the density function from a to b. This integral is the area