ECE 434: Random Processes Exam 2, University of Illinois at Urbana-Champaign, Spring 2003, Exams of Electrical and Electronics Engineering

Download ECE 434: Random Processes Exam 2, University of Illinois at Urbana-Champaign, Spring 2003 and more Exams Electrical and Electronics Engineering in PDF only on Docsity! University of Illinois at Urbana-Champaign ECE 434: Random Processes Spring 2003 Exam 2 Monday, April 14, 2003 Name: • You have 75 minutes for this exam. The exam is closed book and closed note, except that you may consult both sides of two sheets of notes, typed in font size 10 or equivalent handwriting size. • Calculators, laptop computers, Palm Pilots, two-way e-mail pagers, etc. may not be used. • Write your answers in the spaces provided. • Please show all of your work. Answers without appropriate justification will receive very little credit. If you need extra space, use the back of the previous page. Score: 1. (9 pts.) 2. (8 pts.) 3. (6 pts.) 4. (8 pts.) 5. (9 pts.) Total: (40 pts.) 1 Problem 1 (9 points) Let X be a stationary continuous-time Markov process with the transition rate diagram shown. (a) Write down the Q matrix and find the first-order probability distribution π. (So π is a probability vector, representing the distribution of Xt for each t.) (b) Let X2 denote the process X2 = (X2t : t ∈ IR). Is X2 a Markov process? Justify your answer. (c) Is (Xt : t ≥ 0) a martingale? Justify your answer. 2 Problem 4 (8 points) Let Z = (Zt : t ∈ IR) be a mean zero, stationary Gaussian process with RZ(τ) = e−|τ | and let V = ∫∞ 0 e −tZtdt. (a) Find E[V 2]. (b) Find P [V ≥ 3]. 5 Problem 5 (9 points) Let Bk : k ≥ 0 be a sequence of independent random variables with P [Bk = 1] = P [Bk = 0] = P [Bk = −1] = 13 for all k. Let Y = (Yt : t ≥ 0) be the continuous time random process such that • Y has continous sample paths • Y0 = 0 • On each interval of the form [k, k + 1], Y has slope Bk (a) Is Y a Markov process? Justify your answer. (b) Is Y m.s. continuous? Justify your answer. (c) Is Y m.s. differentiable? Justify your answer. 6