Correlation is an important measure of the degree to which two or more random variables have some relationship. For example, if X and Y are correlated, then outcomes of X might be used to predict outcomes of Y. Correlation also means that if X is conditioned on Y (or vice versa), then the conditional pdf of X given Y differs from the marginal pdf. We observed this previously in Chapter 4 for the bivariate Gaussian pdf in Example 4.5. In this section, we discuss normalized measures of correlation that can also be used to generate random variables with a specified degree of correlation.

Definition: Pearson's Correlation Pearson's correlation coefficient for random variables X and Y is

(5.231) Numbered Display Equation

Generally, when a correlation coefficient is mentioned, it refers to Pearson's (and not the other coefficients described below).

Due to the normalization, is dimensionless and bounded, which allows for more meaningful interpretations of the degree of correlation between two random variables.

Theorem 5.8 The correlation coefficient is bounded as follows:

(5.232) Numbered Display Equation

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