DSC 155 & MATH 182: “WINTER ” 2020 TAKE HOME MIDTERM E XAM Available Monday, February 10 Due Tuesday, February 11 Turn in the exam on Gradescope before midnight. You may use whatever text / online resources you like, provided you cite them in your solutions. Do not consult any other humans about this exam. 1. Let A 2 M m⇥n and B 2 M n⇥p be matrices. (a) (3 points) Show that Nullspace(B) ✓ Nullspace(AB). (b) (3 points) Conclude that rank(AB) rank(B).[Hint: Use the rank-nullity theo- rem.] (c) (2 points) Show also that rank(AB) rank(A).[Hint: You may use the fact that rank(C > ) = rank(C ) for any matrix C .] 2. (8 points) Let X be a random variable, whose moment generating function M X (t)= E(e tX ) is < 1 for some fixed t 2 R. We wish to estimate the moment generating function’s value ✓ = M X (t) from data x 1 ,...,x N modeled as i.i.d. samples of X . We consider the following two estimators: T N (x 1 ,...,x N )= 1 N N X j =1 e tx j , U N (x 1 ,...,x N ) = exp t N N X j =1 x j ! . Determine whether T N and U N are (a) unbiased, and (b) consistent estimators for ✓. 3. (6 points) Let {x 1 ,..., x N } be data in R m with sample covariance matrix C. Suppose u is an eigenvector of C, with eigenvalue σ 2 . Let P be the orthogonal projection onto span{u}. Let y j = P x j . Show that the sample variance of the data {y 1 ,..., y N } is precisely σ 2 . 4. Consider the following Gaussian noise model. Let d N m be positive integers. Let Z 1 ,..., Z N be m-dimensional random vectors with normal distribution N (0, υ 2 I m ), and let X j = μ + Qβ j + Z j for 1 j N , where υ 2 > 0, μ 2 R m , β j 2 R d , and Q 2 M m⇥d , Q > Q = I d are unknown parameters. (a) (4 points) Show that the MLE for (μ, Q, β 1 ,..., β N ) is precisely PCA. [Hint: Do not attempt to compute the MLE. Just set up the maximization problem that de- termines the MLE, and show that it leads to the same problem whose solution is PCA.] (b) (4 points) Find the MLE for the unknown common variance parameter υ 2 ; express it in terms of the singular values of the sample covariance matrix of the given data. -