Covariance from variances of sum and difference

Question

For random variables $X$ and $Y$ , the variance of $X-Y$ is $10$ and the variance of $X+Y$ is $18$ . What is $\text{Cov}(X,Y)$ ?

Short answer

$2$ . Expanding both variances, the $\text{Var}(X)$ and $\text{Var}(Y)$ terms cancel in the difference: $\text{Var}(X+Y)-\text{Var}(X-Y)=4\,\text{Cov}(X,Y)$ , so $\text{Cov}(X,Y)=8/4=2$ .

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Lessons behind this question

Covariance and Correlation
Value-at-Risk: Definition, Properties, and the Three Computation Methods
Sample Spaces, Events, and the Probability Axioms

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