TABLE OF CONTENTS We now develop the precise definition of a random variable. As you might guess, a technical condition for random variables is required because of certain subtle pathological problems that have to do with the ability to determine probabilities for the random variable. To arrive at the precise definition, we start with the informal definition of a random variable that we have already given and then show the inevitable difficulty that results without the technical condition. We have informally defined a random variable as being a function on a sample space. Suppose we have a probability space ( ?, 
, P). Let f : ? ? 
be a function mapping the same space into the real line so that f is a candidate random variable. Since the selection of the original sample point ? is random, that is, governed by a probability measure, the output of our measurement of random variable f( ?) should also be random. That is, we should be able to find the probability of an output event such as the event the outcome of the random variable f was between a and b, that is, the event F ? given by F = ( a, b). Observe that there are two different kinds of events being considered here:
output events or members of the event space of the range or range space of the random variable, that is,...
