Download ECE 413 Exam II: Probability for Engineering - Univ. of Illinois at Urbana-Champaign and more Study notes Statistics in PDF only on Docsity! University of Illinois at Urbana-Champaign ECE 413: Probability with Engineering Applications Spring 2007 Exam II Monday, April 9, 2007 Name: • You have 60 minutes for this exam. The exam is closed book and closed note, except you may consult both sides of two 8.5′′ × 11′′ sheets of notes in ten point font size or larger, 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. (24 pts.) 2. (20 pts.) 3. (20 pts.) 4. (16 pts.) 5. (20 pts.) Total: (100 pts.) 1 1. [24 points] A random variable X is observed. Under hypothesis H0, X has the Poisson distribution with mean λ0 = 1. Under hypothesis H1, X has the Poisson distribution with mean λ1 = 3. (a) [8 points] Describe the maximum likelihood decision rule for selecting one of the hypotheses, given that X = n is observed. Be as explicit as possible. (Hint: e ≈ 2.7, e2 ≈ 7.4, e3 ≈ 20, e4 ≈ 54, e5 ≈ 148.) (b) [8 points] For what values of the prior probability π1 (if any) does the MAP rule select hypothesis H1 for all n ≥ 0? (c) [8 points] Suppose that two observations are available, instead of one. Suppose these obser- vations are made under the same hypothesis, and that, given which hypothesis is true, the two observations are conditionally independent, and each has the same distribution as X above. Let n1 and n2 denote the two observations. Finally, suppose the prior probabilities assigned to the hypotheses are π0 = 0.8 and π1 = 0.2. Identify all pairs (n1, n2) such that the MAP rule decides that H1 is true. Be as explicit as possible. 2 4. [16 points] Suppose X is a random variable with mean 10 and variance 3. Find the numerical value of P{X < 10 − √ 3} (or, nearly equivalently, P{X < 8.27}) for the following two choices of distribution type: (a) [8 points] Assuming X is a Gaussian random variable. (b) [8 points] Assuming X is a uniform random variable. 5 5. [20 points] Suppose Z is a uniform random variable on the interval [−2, 4]. (a) [10 points] Find E[|Z|]. (b) [10 points] Give the pdf of Y = |Z| in equation form (specify it for all v, −∞ < v <∞) and sketch the pdf. 6
