Largest possible covariance

EasyRandom Variables Statistics~3m

Two random variables satisfy $\operatorname{Var}(X_1) = 24$ and $\operatorname{Var}(X_2) = 6$ . Over all joint distributions consistent with these variances, what is the maximum possible value of $\operatorname{Cov}(X_1, X_2)$ ?

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Write $\operatorname{Cov}=\rho\,\sigma_{X_1}\sigma_{X_2}$ and recall $|\rho|\le 1$ .
The max is $\sqrt{\operatorname{Var}(X_1)\operatorname{Var}(X_2)}$ , at $\rho=1$ .

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Largest possible covariance

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