Discussion
For two random variables \(x\) and \(y\) covariance between \(2x\) and \(y\), \(\mathrm{cov}(2x, y) = 4\). Hence, \(\mathrm{cov}(5x - 2, 2y - 5)\) is :
Answer: D
\(\mathrm{cov}(5x - 2, 2y - 5) =\) 20.
Covariance ignores added constants and is bilinear in scale. From \(\mathrm{cov}(2x, y) = 2\,\mathrm{cov}(x, y) = 4\), \(\mathrm{cov}(x, y) = 2\). Then
\[ \begin{aligned} &\mathrm{cov}(5x - 2, 2y - 5) \\ &= 5 \cdot 2 \cdot \mathrm{cov}(x, y) = 10 \cdot 2 = 20 \end{aligned} \]
40 multiplies the given value 4 by 10 without first dividing by 2, 10 forgets that \(\mathrm{cov}(x, y)\) is 2 rather than 1, and 0 would wrongly treat the shifts as removing the relationship.
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