3. Consider a principal components analysis of a random vector X. Let 11 > 12 > ... >lp denote the eigenvalues of the covariance matrix of X and let e1, ..., ey denote the corre- sponding eigenvectors. Suppose that instead of observing X we observe Y = X + where e is a mean-zero random vector with covariance matrix of the form oI, and Cov(, X) = 0. (a) Do you expect that the principal components based on the covariance matrix of Y will be the same as the principal components based on the covariance matrix of X. Why or why not? (b) Suppose that two principal components adequately summarize the variation in X. Will two principal components be adequate for summarizing the variation in Y? Why or why not? In answering this question, you can either give a mathematical argument or give an informal description of what you expect to be true and why. 3. Consider a principal components analysis of a random vector X. Let 11 > 12 > ... >lp denote the eigenvalues of the covariance matrix of X and let e1, ..., ey denote the corre- sponding eigenvectors. Suppose that instead of observing X we observe Y = X + where e is a mean-zero random vector with covariance matrix of the form oI, and Cov(, X) = 0. (a) Do you expect that the principal components based on the covariance matrix of Y will be the same as the principal components based on the covariance matrix of X. Why or why not? (b) Suppose that two principal components adequately summarize the variation in X. Will two principal components be adequate for summarizing the variation in Y? Why or why not? In answering this question, you can either give a mathematical argument or give an informal description of what you expect to be true and why