Variance of the gap between two linked quantities

Question

Two related quantities have variances \mathrm{Var}(X) = 10 and \mathrm{Var}(Y) = 6, with covariance \mathrm{Cov}(X, Y) = 4. You track their gap, D = X - Y. What is the variance of the gap?

Accepted Answer

For a difference, the rule is almost the same as for a sum, except the covariance term now gets subtracted: \mathrm{Var}(X - Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) - 2\,\mathrm{Cov}(X, Y). In words: add the two variances, then subtract twice the covariance. The subtraction makes sense because when two quantities move together, their gap stays steadier, so positive covariance shrinks the spread of the difference. Apply it. \mathrm{Var}(D) = 10 + 6 - 2(4) = 16 - 8 = 8. The standard deviation is \sqrt{8} \approx 2.83. Because the two quantities move together somewhat, the gap between them is cal…

Variance of the gap between two linked quantities

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