Applied Statistics and Probability for Engineers

54 CHAPTER 2 PROBABILITY

Sometimes a measurement (such as current in a copper wire or length of a machined part)

can assume any value in an interval of real numbers (at least theoretically). Then arbitrary pre-

cision in the measurement is possible. Of course, in practice, we might round off to the nearest

tenth or hundredth of a unit. The random variable that represents this measurement is said to

be a continuousrandom variable. The range of the random variable includes all values in an

interval of real numbers; that is, the range can be thought of as a continuum.

In other experiments, we might record a count such as the number of transmitted bits that

are received in error. Then the measurement is limited to integers. Or we might record that a

proportion such as 0.0042 of the 10,000 transmitted bits were received in error. Then the

measurement is fractional, but it is still limited to discrete points on the real line. Whenever

the measurement is limited to discrete points on the real line, the random variable is said to be

a discreterandom variable.

A random variableis a function that assigns a real number to each outcome in the

sample space of a random experiment.

A random variable is denoted by an uppercase letter such as X. After an experi-

ment is conducted, the measured value of the random variable is denoted by a low-

ercase letter such as x 70 milliamperes.

Definition

A discreterandom variable is a random variable with a finite (or countably infinite)

range.

A continuousrandom variable is a random variable with an interval (either finite or

infinite) of real numbers for its range.

Definition

In some cases, the random variable Xis actually discrete but, because the range of possible

values is so large, it might be more convenient to analyze Xas a continuous random variable. For

example, suppose that current measurements are read from a digital instrument that displays the

current to the nearest one-hundredth of a milliampere. Because the possible measurements are

limited, the random variable is discrete. However, it might be a more convenient, simple approx-

imation to assume that the current measurements are values of a continuous random variable.

Examples of continuous random variables:

electrical current, length, pressure, temperature, time, voltage, weight

Examples of discreterandom variables:

number of scratches on a surface, proportion of defective parts among 1000

tested, number of transmitted bits received in error.

Examples of

Random

Variables

EXERCISES FOR SECTION 2-8

2-100. Decide whether a discrete or continuous random

variable is the best model for each of the following vari-

ables:

(a) The time until a projectile returns to earth.

(b) The number of times a transistor in a computer memory

changes state in one operation.

(c) The volume of gasoline that is lost to evaporation during

the filling of a gas tank.

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