2 Random Variables
2.1 DEFINITION OF A RANDOM VARIABLE
An experiment 
is specified by the three tuple (S, 
, 
(.)) where S is a finite, countable, or noncountable set called the sample space, is a Borel field specifying a set of events, and (.) is a probability measure allowing calculation of probabilities of all events. Using an underlying experiment a random variable X(ζ) is defined as a real-valued function on S that satisfies the following: (a) {ζ : X(ζ) ≤ x} is a member of for all x, which guarantees the existence of the cumulative distribution function, and (b) the probabilities of the events {ζ : X(ζ) = +∞} and {ζ : X(ζ) = −∞} are both zero. This means that the function is not allowed to be + or − infinite with a nonzero probability.
EXAMPLE 2.1
is specified by (S, , (.)) where
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