ECE 534: Random Processes Exam 2, University of Illinois at Urbana-Champaign, Fall 2008, Exams of Electrical and Electronics Engineering

Download ECE 534: Random Processes Exam 2, University of Illinois at Urbana-Champaign, Fall 2008 and more Exams Electrical and Electronics Engineering in PDF only on Docsity! University of Illinois at Urbana-Champaign ECE 534: Random Processes Fall 2008 Exam 2 Monday, November 17, 2008 Name: • You have 75 minutes for this exam. The exam is closed book and closed note, except you may consult both sides of two 8.5′′ × 11′′ sheet of notes. • 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. (12 pts.) 2. (12 pts.) 3. (16 pts.) Total: (40 pts.) 1 Problem 1 (12 points) Suppose a = (a0, a1, a2, a3) is a probability vector to be estimated by observing Y = (Y1, T2, · · · , YT ). Assume Y1, . . . , YT are independent, and P{Yt = i} = ai for 1 ≤ t ≤ T and i ∈ {0, 1, 2, 3}. (a)(6 points) Determine the maximum likelihood estimate, âML(y), given a particular observation y = (y1, . . . , yT ). Justify your answer. (b)(6 points) Suppose in addition to the above that a has the form ai = ( 3 i ) qi(1 − q)3−i, or in other words, that a is a binomial pmf with parameters (3,q). Determine the maximum likelihood estimate, q̂ML(y), given a particular observation y = (y1, . . . , yT ). Justify your answer. 2