Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

2.14 CONTINUOUS SAMPLE SPACES

For a continuous sample space with an uncountable number of outcomes, the situation is quite different because it is not possible to assign nonzero probabilities to individual sample points in such a way that they sum to 1. For example, consider an experiment where the outcomes are nonnegative real numbers . It is clear that for any two numbers there is always an infinity of numbers in between (which is not the case for an experiment with a countable number of outcomes). If we attempted to assign probabilities to individual numbers, we would realize very quickly that those probabilities cannot be distributed among all the numbers in and still satisfy the three axioms. The probability of a single number in an experiment with continuous outcomes must be zero. Although this result may seem to be counterintuitive at first because individual outcomes obviously do occur in a continuous experiment, some reflection shows that this must be the case. For example, the probability that occurs is obviously “very unlikely”; it almost surely will not occur, which is another way of saying that .

Probability assignments for continuous experiments are handled by considering ...