Random Variables, Probability Distributions, and Characteristic Functions

The outcome of a random experiment may be any one of a great variety of objects: heads and tails in tossing a coin; the integers 1, 2, 3, 4, 5, 6 when a die is tossed; red, white, and black balls when a ball is drawn from an urn containing balls of these colors. Thus the outcomes are not always real numbers. In most practical applications the people deal with random variables that can by and large be divided into two classes: discrete and continuous. However, in many cases, the outcomes of a random experiment cannot be fully expressed by a single (univariate) random variable, and the people need to consider several such variables to give a clear expression of the outcome. The correlation coefficient gives the reader a numerical measure of the linear association between X and Y.