The largest a covariance can possibly be

Question

Two quantities have variances 9 and 4. A colleague claims their covariance is 10. What is the largest possible covariance for these two quantities, and is the claim of 10 possible?

Accepted Answer

The key fact is that correlation can never leave the range from -1 to +1. Since correlation is the covariance divided by the two standard deviations, ho = \frac{\mathrm{Cov}(X, Y)}{\mathrm{SD}(X)\,\mathrm{SD}(Y)}, and ho can be at most 1 in size, the covariance itself can be at most the product of the two standard deviations: \mathrm{Cov}(X, Y) \le \mathrm{SD}(X)\,\mathrm{SD}(Y). In words: the covariance can never be larger in size than the two spreads multiplied together. Compute the cap. \mathrm{SD}(X) = \sqrt{9} = 3, \qquad \mathrm{SD}(Y) = \sqrt{4} = 2, \qquad 	ext{cap} = 3 	imes 2 …

The largest a covariance can possibly be

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