Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 2 / 3F / Solution

For random variables with finite second moments, their covariance isThe Cauchy-Schwarz inequality ensures the product is integrable under this hypothesis. More generally the definition works whenever the displayed expectations exist. Assertions about arbitrary random variables below are understood on this domain: without suitable moments a covariance need not be defined at all.