Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 2 / 3F (Probability)

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.

Linearity of expectation gives $\mathbb{E}(X+Y)=\mathbb{E}X+\mathbb{E}Y$. Expand the centered product and use linearity of expectation again:Thus the assertion is true whenever the covariances are defined. Neither independence nor identical distributions are required.

The covariance is symmetric, because multiplication of real random variables is commutative. Expanding with the bilinearity just proved givesIdentical probability distributions imply equal second moments and equal means, hence equal variances. Therefore the assertion is true, withThe cancellation shows exactly why no assumption of independence is needed.