Let \(X_{1}\) and \(X_{2}\) be two positively correlated random variables, both with variance 1 .

a. The first principal component, \(P C_{1}\), is the linear combination of \(X_{1}\) and \(X_{2}\) that maximizes \(\operatorname{var}\left(w_{1} X_{1}+w_{2} X_{2}\right)\), where \(w_{1}^{2}+w_{2}^{2}=1\). Show that \(P C_{1}=\left(X_{1}+X_{2}\right) / \sqrt{2}\). (Hint: First derive an expression for \(\operatorname{var}\left(w_{1} X_{1}+w_{2} X_{2}\right)\) as a function of \(w_{1}\) and \(w_{2}\).)

b. The second principal component is \(P C_{2}=\left(X_{1}-X_{2}\right) / \sqrt{2}\). Show that \(\operatorname{cov}\left(P C_{1}, P C_{2}\right)=0\).

c. Show that \(\operatorname{var}\left(P C_{1}\right)=1+ho\) and \(\operatorname{var}\left(P C_{2}\right)=1-ho\), where \(ho=\operatorname{cor}\left(x_{1}, x_{2}\right)\).