3.3.1 Distributions

Suppose we have a probability space ( ?, , P) with a random variable, X, defined on the space. The random variable X takes values on its range space which is some subset A of (possibly A = ). The range space A of a random variable is often called the alphabet of the random variable. As we have seen, since X is a random variable, we know that all subsets of ? of the form X ^?1( F) = { ? : X( ?) ? F}, with F ? ( A), must be members of by definition. Thus the set function P _X defined by

is well defined and assigns probabilities to output events involving the random variable in terms of the original probability of input events in the original experiment. The three written forms in Equation (3.22) are all read as Pr( X ? F) or the probability that the random variable X takes on a value in F. Furthermore, since inverse images preserve all set-theoretic operations (see Problem A.12), P _X satisfies the axioms of probability as a probability measure on ( A, ( A)) it is nonnegative, P _X( A) = 1, and it is countably additive. Thus P _X is a probability measure on the measurable space...