The probability distribution of a random variable specifies the chance that the variable takes a value in any subset of the real numbers. (The subsets have to satisfy some technical conditions that are not important for this course.) The probability distribution of a random variable is completely characterized by the cumulative probability distribution function; the terms sometimes are used synonymously. The probability distribution of a discrete random variable can be characterized by the chance that the random variable takes each of its possible values. For example, the probability distribution of the total number of spots S showing on the roll of two fair dice can be written as a table:

The probability distribution of a continuous random variable can be characterized by its probability density function.