Why does it matter? We have listed the properties of the CDF for a very important reason. Every random variable has a CDF with the properties listed. Also the reverse is true. Given any CDF F with the properties listed, we can find a random variable X such that F is the CDF of X. This is called a characterization theorem. It shows that the set of random variables is one-to-one mapped to the CDF functions with the listed properties. This allows us to use CDF's to represent random variables, and assures us mathematically that all relevant information about the random variable is contained in the CDF>