Page 39 - Applied statistics and probability for engineers

Section 2-1/Sample Spaces and Events 17

Controlled
variables

Physical system
Input System Output
Measurements Analysis

Model Noise
variables
FIGURE 2-1 Continuous iteration between FIGURE 2-2 Noise variables affect the
model and physical system. transformation of inputs to outputs.

Random
Experiment An experiment that can result in different outcomes, even though it is repeated in the
same manner every time, is called a random experiment.

For the example of measuring current in a copper wire, our model for the system might
simply be Ohm’s law. Because of uncontrollable inputs, variations in measurements of cur-
rent are expected. Ohm’s law might be a suitable approximation. However, if the variations
are large relative to the intended use of the device under study, we might need to extend our
model to include the variation. See Fig. 2-3.
As another example, in the design of a communication system, such as a computer or voice
communication network, the information capacity available to serve individuals using the net-
work is an important design consideration. For voice communication, suficient external lines
need to be available to meet the requirements of a business. Assuming each line can carry only
a single conversation, how many lines should be purchased? If too few lines are purchased, calls
can be delayed or lost. The purchase of too many lines increases costs. Increasingly, design and
product development is required to meet customer requirements at a competitive cost.
In the design of the voice communication system, a model is needed for the number of calls
and the duration of calls. Even knowing that, on average, calls occur every ive minutes and
that they last ive minutes is not suficient. If calls arrived precisely at ive-minute intervals
and lasted for precisely ive minutes, one phone line would be suficient. However, the slight-
est variation in call number or duration would result in some calls being blocked by others.
See Fig. 2-4. A system designed without considering variation will be woefully inadequate for
practical use. Our model for the number and duration of calls needs to include variation as an
integral component.

2-1.2 SAMPLE SPACES

To model and analyze a random experiment, we must understand the set of possible outcomes
from the experiment. In this introduction to probability, we use the basic concepts of sets and
operations on sets. It is assumed that the reader is familiar with these topics.

Sample Space
The set of all possible outcomes of a random experiment is called the sample space
of the experiment. The sample space is denoted as S.

A sample space is often deined based on the objectives of the analysis. The following example
illustrates several alternatives.