Exercise 6.6: Variance and Positive Semidefiniteness

chapter 6

Since \(\E [(Q - \E[Q])^2] = \E [Q^2] - (\E[Q])^2\), and \((Q - \E[Q])^2\) is positive semidefinite, the result follows by the fact that positive semidefinite matrices form a closed convex cone (see, e.g., theorem 3.1 here), and the expectation can be expressed as a limit of convex combinations.