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Appendix I: Review of SomeProbability
and Statistics Concepts
Probability Concepts
Probability begins with the ideas of “sample space” and “experiment.” An
experiment is the observation of some phenomenon whose result cannot
beperfectly predicted a priori. A sample space is the collection of all possible
results (called outcomes) from an experiment. Thus, an experiment can
bethought of as the observation of a result taken from a sample space. These
circular denitions may be a little annoying and somewhat bafing,but they
are easily illustrated. If the experiment is to observe which face of a six-sided
die lands up after throwing it across a gaming table, then the sample space
consists of six elements, namely, the array of one, two, three, four, ve, or
six dots, as they are typically arrayed on the faces of a six-sided die. Sample
spaces need not be so discrete or nite; they can be continuous and in-
nite, in that they can have an innite number of outcomes. For example, if a
sample space consists of all possible initial voltages generated by LiI batteries
made in a battery manufacturing plant, then it would have an innite (albeit
bounded) number of possible outcomes.
A random variable is a mapping from a sample space into (usually) some
subset of the real numbers (possible over the entire real line). Think of the
random variable as a “measurement” taken after the experiment is per-
formed. Thus, the number of dots in the array showing after the die is cast, or
the voltage as measured by a volt meter, would be random variables. There
are two basic classes of random variables, discrete and continuous. Discrete
random variables are mapped from the sample space to a subset of integers,
and continuous random variables are mapped to subsets of real numbers.
The die example is discrete, and the voltage example is continuous.
Every random variable has a probability distribution function that
describes the chances of observing particular ranges of values for the random
variable. In the case of discrete random variables, it also makes sense to talk
about the probability of an experiment resulting in a particular value, for
example, the probability that the number of dots in the die array showing is
four. For continuous variables, it makes sense to talk about the probability of
obtaining a value in a “small” range, but the probability of obtaining a par-
ticular value is zero. This is not to say that particular values of continuous