Page 71 - Applied Statistics And Probability For Engineers

c02.qxd 5/10/02 1:07 PM Page 54 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

54 CHAPTER 2 PROBABILITY

Definition
A random variable is 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.

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 continuous random 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 discrete random variable.

Definition
A discrete random variable is a random variable with a finite (or countably infinite)
range.
A continuous random variable is a random variable with an interval (either finite or
infinite) of real numbers for its range.

In some cases, the random variable X is actually discrete but, because the range of possible
values is so large, it might be more convenient to analyze X as 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
Random Examples of continuous random variables:
Variables
electrical current, length, pressure, temperature, time, voltage, weight
Examples of discrete random variables:
number of scratches on a surface, proportion of defective parts among 1000
tested, number of transmitted bits received in error.

EXERCISES FOR SECTION 2-8
2-100. Decide whether a discrete or continuous random (b) The number of times a transistor in a computer memory
variable is the best model for each of the following vari- changes state in one operation.
ables: (c) The volume of gasoline that is lost to evaporation during
(a) The time until a projectile returns to earth. the filling of a gas tank.