Dependence, Correlation, and Conditional Expectation
So far, when dealing with a sequence of random variables, we always assumed that they were independent. In this chapter, at last, we investigate the issue of dependence. To get the basic intuition, consider the hypothetical demand data in Table 8.1.
Can we say that the two random variables D1 and D2 are independent? Looking at the raw data may be a bit confusing, but things get definitely clearer if we consider the sample means, and . We observe that whenever D1 is above average, D2 tends to be, too; vice versa, whenever D1 is below average, D2 tends to be, too. Hence, we have a sort of concordance between the two random variables, which should not be there if the two variables were independent. Capturing this concordance leads us to the definition covariance and correlation, which is the main purpose of this chapter.
Before doing so, we introduce the formal concepts of joint and marginal distributions in Section 8.1. The mathematics here is a bit more complicated than elsewhere, and the section can be skipped by those who just wish an intuitive understanding. In Section ...