Page 54 - Fundamentals of Probability and Statistics for Engineers

3

Random Variables and Probability

Distributions

We have mentioned that our interest in the study of a random phenomenon is in the
statements we can make concerning the events that can occur, and these statements
are made based on probabilities assigned to simple outcomes. Basic concepts have
been developed in Chapter 2, but a systematic and unified procedure is needed to
facilitate making these statements, which can be quite complex. One of the immedi-
ate steps that can be taken in this unifying attempt is to require that each of the
possible outcomes of a random experiment be represented by a real number. In this
way, when the experiment is performed, each outcome is identified by its assigned
real number rather than by its physical description. For example, when the possible
outcomes of a random experiment consist of success and failure, we arbitrarily assign
the number one to the event ‘success’ and the number zero to the event ‘failure’. The
g
f
associated sample space has now 1, 0 as its sample points instead of success and
failure, and the statement ‘the outcome is 1’ means ‘the outcome is success’.
This procedure not only permits us to replace a sample space of arbitrary
elements by a new sample space having only real numbers as its elements but
also enables us to use arithmetic means for probability calculations. Further-
more, most problems in science and engineering deal with quantitative meas-
ures. Consequently, sample spaces associated with many random experiments
of interest are already themselves sets of real numbers. The real-number assign-
ment procedure is thus a natural unifying agent. On this basis, we may intro-
X
duce a variable , which is used to represent real numbers, the values of which
are determined by the outcomes of a random experiment. This leads to the
notion of a random variable, which is defined more precisely below.

3.1 RANDOM VARIABLES

Consider a random experiment to which the outcomes are elements of sample
space S in the underlying probability space. In order to construct a model for

Fundamentals of Probability and Statistics for Engineers T.T. Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)

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