Discussion
Let \(X \sim N_{4}(\mu, \Sigma)\) where eigen values of \(\Sigma\) are given by \(\lambda_{1} = 6\), \(\lambda_{2} = 4\), \(\lambda_{3} = 3\), \(\lambda_{4} = 2\). Let \(Y_{i}\) be the principal component corresponding to \(\lambda_{i}\), \(i = 1, 2, 3, 4\). Then which of the following statements is not correct ?
Answer: B
The statement that is not correct is \(Y_{1}, Y_{2}, Y_{3}, Y_{4}\) are identically normally distributed.
The principal components are \(Y_{i} = e_{i}'X\) for orthonormal eigenvectors \(e_{i}\) of \(\Sigma\). Each is normal with variance \(\lambda_{i}\), and their covariances are 0; jointly normal and uncorrelated means independent, so (A) is true.
But the variances 6, 4, 3, 2 differ, so the components are not identically distributed. The others are true: the first two explain \((6 + 4)/15 \approx 66.7\%\) of the total variance 15, more than 60%, and \(Y_{4}\) has the smallest eigenvalue, 2.
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