STA 135 Sample Exam I
Disclaimer : This is just a practice exam. If something is not cover here that does not mean it won’t be covered during the
For each of the following questions indicate true or false, then explain your answer. You may use examples to illustrate your
(I) Principal component analysis (PCA) can be used with variables of any mathematical types: quantitative, qualitative,
or a mixture of these types.
(II) If the covariance of two random variable X and Y equals 0, then they are independent.
(III) In multivariate analysis, we always need to scale the variable before performing PCA ?
(IV) The mahalanobis distance is always different to the traditional ||.||2 norm.
Work out the following problems. Show your work.
1. Let X and Y be jointly biviriate normal with V ar(X) = V ar(Y ). Show that the two random variables X + Y and
X − Y are independent.
2. Let X1, X2 and X3 be independent and identically distributed 3× 1 random vecotrs with
µ = [3,−1, 1]T and
3 −1 1−1 1 0
1 0 2
(a) Find the mean and variance of Y1 = aTX1 of the three components of X1 where a = [a1, a2, a3]T .
(b) Find the distribution of 2X1 −X2 +X3. A complete formula will suffice.
3. Let Σ1 and Σ2 two population covariance matrices defined as follows,
3 0 00 1 0
0 0 2
1 0 −10 1 0
−1 0 1
(a) Compute Σ−11
(b) Is Σ2 is positive definite or semidefinite ? Justify your answer.
(c) Provide the spectra decomposition of Σ1. Clearly specify the different parts.
(d) Should we apply principal components on any of these matrices ? Justify your answer.
(e) Determine the population principal components Y1, Y2 and Y3 for the covariance matrix Σ2. Use the eigenvalues
λ1 = 2, λ2 = 1, λ3 = 0.
(f) Also, calculate the proportion of the total population variance explained by the first two principal components of
(g) Following up on the previous question, how many principal components are needed to explain more than 90% of
the variability of the data ?
(h) In the practical sense, is there any reason why any two principal components should not be orthogonal ?
4. Prove that every eigenvalues of a k× k positive definite matrix A is positive. Hint: Consider the following relationship
Au = λu where u is an eigenvalue with corresponding eigenvector λ.