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QUESTION BOOKLET
EE 126 Spring 2006 Midterm
Thursday, April 13, 11:10-12:30pm
DO NOT OPEN THIS QUESTION
BOOKLET UNTIL YOU ARE TOLD TO
DO SO
- You have 80 minutes to complete the midterm.
- The midterm consists of three problems, provided in the question booklet (THIS BOOKLET), that are in no particular order of difficulty.
- Write your solution to each problem in the space provided in the solution booklet (THE OTHER BOOKLET). Try to be neat! If we can’t read it, we can’t grade it.
- You may give an answer in the form of an arithmetic expression (sums, products, ratios, factorials) that could be evaluated using a calculator. Expressions like
3
or ∑ 5 k=0(1/2) k (^) are also fine.
- A correct answer does not guarantee full credit and a wrong answer does not guarantee loss of credit. You should concisely explain your reasoning and show all relevant work. The grade on each problem is based on our judgment of your understanding as reflected by what you have written.
- This is a closed-book exam except for two sheets of handwritten notes (one 8. 5 × 11 page, both sides OR two pages, single side only), plus a calculator.
Problem 1: (12 points)
Diana the Daredevil is trying to break the world land-speed record using her rocket- powered motorcycle. In order to do so, she needs to cover more than 500 yards in 10 seconds. If her motorcycle has Z pounds of rocket fuel to start, then it travels a distance of X = 50
Z yards in 10 seconds.
(a)(5 points) Suppose that the amount of rocket fuel Z is a random variable uniformly distributed over [50, 150]. Compute the PDF and CDF of the distance X.
(b)(2 points) Compute the probability that Diana breaks the world record on any given trial.
(c)(5 points) Now suppose that Diana is allowed to take a total of n trials in order to try and break the world record. (Here n is some fixed positive integer: i.e., n ∈ { 1 , 2 , 3 ,.. .}). The amount of rocket fuel at the start is independent from trial to trial. What is the smallest possible integer n such that Diana has a better than 90% chance of breaking the world record over the n trials?
Problem 3: (14 points) George the Gambler is a good poker player: every round that he plays, he wins a random amount Z of money, distributed as Z ∼ N (μ, σ^2 ) with μ > 0. Every night, George goes to the local poker club, and plays T = 1 + V rounds of poker, where V is a Poisson random variable with parameter λ = 5. Let X be the total amount of money that George wins in a given night.
(a)(2 points) In the absence of any further information, what is the best estimate of X? (Here “best” is measured by the minimum mean-squared error.)
(b)(3 points) Suppose that you observe that George plays T = t rounds of poker (where t is some positive integer). Now what is the best estimate of X (again measured in terms of minimim mean-squared error)?
(c)(5 points) Now suppose that by peeking into George’s wallet at the end of the night, you make a noisy observation Y of the amount of money X that he won—say of the form
Y = X + W
where W ∼ N (0, 1) is Gaussian noise independent of X. Compute the Bayes’ least squares estimate of X based on observing T = t and Y = y. Also compute the linear least squares estimate (LLSE) of X based on observing T = t and Y = y, and the error variance of the LLSE.
(d)(4 points) Now suppose that you observe that {T ≤ 2 } and Y = y. Compute the Bayes’ least squares estimate of X based on this information.
