Maximize Covariance from Known Variances

Random variables $X$ and $Y$ have finite variances $\mathrm{Var}(X) = 3$ and $\mathrm{Var}(Y) = 27$. Determine the largest possible value of $\mathrm{Cov}(X, Y)$. Justify the upper bound and explain why it can be reached.

Constraints & Assumptions

• Independence is not assumed.
• The means are arbitrary and do not affect the bound on covariance.

Clarifying Questions to Ask

• Is the target the maximum signed covariance or the maximum absolute covariance?
• Are any jointly distributed random variables permitted as long as the stated variances are satisfied?

Hint — Center the variables: Use an inner-product inequality on X - E[X] and Y - E[Y].

What a Strong Answer Covers

• The Cauchy–Schwarz inequality for covariance.
• Correct substitution of the two variances.
• An equality condition demonstrating attainability.
• The difference between positive maximum and absolute magnitude.

Follow-up Questions

• What is the smallest possible covariance?
• How does the answer change when $X$ and $Y$ are independent?

Overview: Find the maximum possible covariance when two random variables have variances 3 and 27.