Discussion
Let \((X, Y)\) be a random vector with \(Var(X) = Var(Y) = 1\) and \(Cov(X, Y) = -0.5\). For which one of the following values of \(b\), the random variables \(U = bX + Y\) and \(V = X + bY\) are uncorrelated?
Answer: A
The value is \(b = 2 - \sqrt{3}\).
Expand the covariance using bilinearity:
\[ \begin{aligned} Cov(U, V) &= b\,Var(X) + (b^{2} + 1)Cov(X, Y) + b\,Var(Y) \\ &= 2b - 0.5(b^{2} + 1) \end{aligned} \]
Setting this to zero gives \(b^{2} - 4b + 1 = 0\), so \(b = 2 \pm \sqrt{3}\). Of these only \(2 - \sqrt{3}\) is offered.
\(\sqrt{3} - 2\) is its negative and gives \(Cov(U,V) = 2(\sqrt{3} - 2) - 0.5(8 - 4\sqrt{3}) = 4\sqrt{3} - 8 \neq 0\). Substituting \(b = \sqrt{3}\) gives \(2\sqrt{3} - 2 \neq 0\), and \(b = 2\) gives \(4 - 2.5 = 1.5 \neq 0\).
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