# Maximize Covariance from Known Variances Random variables `X` and `Y` have finite variances `Var(X) = 3` and `Var(Y) = 27`. Determine the maximum possible value of `Cov(X, Y)`. Justify both the upper bound and why it is attainable. ### Constraints & Assumptions - No independence assumption is given. - Means are arbitrary and do not affect the covariance bound. ### Clarifying Questions to Ask - Is the question asking for the maximum signed covariance or maximum absolute covariance? - Are arbitrary jointly distributed random variables allowed as long as the stated variances hold? ```hint Center the variables Apply an inner-product inequality to `X - E[X]` and `Y - E[Y]`. ``` ### What a Strong Answer Covers - The covariance Cauchy-Schwarz inequality. - Correct substitution of the two variances. - An equality condition demonstrating attainability. - The distinction between positive maximum and absolute magnitude. ### Follow-up Questions - What is the minimum possible covariance? - How does the answer change if `X` and `Y` are independent?