Department of EECS - University of California at BerkeleyEECS 226A - Random Processes in Systems - Fall 2006Quiz: 9/7/2006This quiz is to evaluate the background of the students in the class. The results will NOTcount for your grade. Please indicate your name so that I have an updated list of students.SOLUTIONS1. Find (a) the determinant, (b) the eigenvalues, and (c) the eigenvectors of the followingmatrix: (Show your work.)"2 12 3#.(a) The determinant is 2 × 3 − 2 × 1 = 4.(b) To find the eigenvalues of a matrix A, we solve det(A − sI) = 0. Here:det("2 12 3#−"s 00 s#) = det("2 − s 12 3 − s#) = (2 − s)(3 − s) − 2 = s2− 5s + 4.The eigenvalues are λ1= (5 −√25 − 16)/2 = 1 and λ2= (5 +√25 − 16)/2 = 4.(c) To get the eigenvectors vi, we solve Avi= λivi. Thus, for i = 1,"2 12 3#"ab#="ab#.This gives 2a + b = a and 2a + 3b = b. Choosing b = 1, we find a = −1, so v1= [−1, 1]T.For i = 2 we find"2 12 3#"ab#= 4"ab#.Thus, 2a + b = 4a. Choosing b = 1, we get a = 1/2, so that v2= [1/2, 1]T.12. Let A and B be two n × n matrices. Which of the following statements are true?(a) AB = BA — This is FALSE in general.(b) det(AB) = det(BA) — This is TRUE, the common value being det(A)det(B).(c) (AB)T= ATBT— This is FALSE in general. What is true is (AB)T= BTAT.(d) (AB)−1= B−1A−1. — This is true (if the inverses exist) since (AB)(B−1A−1) = I.3. Let X be a random variable uniformly distributed in [0, 1]. Calculate E(Xn) where nis a positive integer. (Show your work.)We find E(Xn) =R10xndx = 1/(n + 1).4. Let X, Y be two i.i.d. random variables geometrically distributed with mean 12.Calculate E[X|X + Y ]. (Show your work.)We have E[X|X + Y ] = E[Y |X + Y ], by symmetry. But E[X|X + Y ] + E[Y |X + Y ] =E[X + Y |X + Y ] = X + Y . Hence, E[X|X + Y ] = (X + Y )/2.5. Let X, Y be two random variables defined on a common probability space. Assumethat E(XY ) = E(X)E(Y ). Is it true that X and Y are independent? Yes or No. — NO;they are just uncorrelated.6. Let X, Y be two i.i.d. random variables that are uniformly distributed in [0, 1].Calculate E(min{X, Y }). Show your work.One hasE(min{X, Y }) =Z∞0P (min{X, Y } > x)dx) =Z10(1 − x)2dx =Z10x2dx