Page 27 - Computational Statistics Handbook with MATLAB

Chapter 2: Probability Concepts 13

• When observing piston ring failures, the sample space is 10,{ , }
where 1 represents a failure and 0 represents a non-failure.
• If we roll a six-sided die and count the number of dots on the face,
then the sample space is 12 34 56, ,,, ,{ . }

The outcomes from random experiments are often represented by an
uppercase variable such as X. This is called a random variable, and its value
is subject to the uncertainty intrinsic to the experiment. Formally, a random
variable is a real-valued function defined on the sample space. As we see in
the remainder of the text, a random variable can take on different values
according to a probability distribution. Using our examples of experiments
from above, a random variable X might represent the failure time of a soft-
ware system or the glucose level of a patient. The observed value of a random
variable X is denoted by a lowercase x. For instance, a random variable X
might represent the number of failures of piston rings in a compressor, and
x = 5 would indicate that we observed 5 piston ring failures.
Random variables can be discrete or continuous. A discrete random vari-
able can take on values from a finite or countably infinite set of numbers.
Examples of discrete random variables are the number of defective parts or
the number of typographical errors on a page. A continuous random variable
is one that can take on values from an interval of real numbers. Examples of
continuous random variables are the inter-arrival times of planes at a run-
way, the average weight of tablets in a pharmaceutical production line or the
average voltage of a power plant at different times.
We cannot list all outcomes from an experiment when we observe a contin-
uous random variable, because there are an infinite number of possibilities.
However, we could specify the interval of values that X can take on. For
example, if the random variable X represents the tensile strength of cement,
then the sample space might be 0 ∞,( ) kg/cm 2 .
An event is a subset of outcomes in the sample space. An event might be
that a piston ring is defective or that the tensile strength of cement is in the
2
range 40 to 50 kg/cm . The probability of an event is usually expressed using
the random variable notation illustrated below.
• Discrete Random Variables: Letting 1 represent a defective piston
ring and letting 0 represent a good piston ring, then the probability
of the event that a piston ring is defective would be written as

(
PX = 1) .

• Continuous Random Variables: Let X denote the tensile strength
of cement. The probability that an observed tensile strength is in
2
the range 40 to 50 kg/cm is expressed as

P 40 kg/cm ≤( 2 X ≤ 50 kg/cm 2 . )

© 2002 by Chapman & Hall/CRC