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August 6th, 2009, 10:41 AM  #1 
Senior Member Joined: Dec 2008 Posts: 206 Thanks: 0  Challenge problems
I have attached two challenge problems in this post . I have just started to have a look at it . If anyone has any idea pls do share with me. 1. Abbey and Beth are friends. So are Carl and Dan. They are having a little dispute about fairness and they ask you to resolve it. Whenever Carl and Dan go out to eat each Saturday, they take strict turns paying the tab. If Carl payed last time, Dan will pay this time and vice versa. But they don't bother about consciously keeping track of money and will order a big dish randomly from the menu and share it equally. Big dish prices are uniformly distributed on [5,10] Abbey and Beth take a different approach to fairness. They too go out to eat and take strict turns paying the bill. But they order small dishes separately. Whenever one of them calculates that she is currently more than $10 behind in her fair share of payments, she will order the salad for $1. Otherwise she randomly picks a dish from the small dish menu where the prices are uniformly distributed on [2.5,5]. Carl and Dan believe that Abbey and Beth are being overly cautious and advise them to trust in the power of averaging to make things work out in the end. Abbey and Beth are skeptical and do not want to risk their friendship. Assume that if at any time the balance of payments becomes skewed by more than M times the maximum price of a dish in one direction, then unconscious feelings of being cheated will manifest and the friendship will grow distant. * Setup a model for AB behavior and CD behavior. * What is the probability that one of Carl and Dan will eventually feel cheated as a function of M? Abbey and Beth? * As M increases, approximately how does the time till friendship strain increase? * What if there is inflation in the system and the price of dishes goes up by five percent per year? * What if there was deflation in the system and the price of dishes dropped by five percent per year? 2. You are looking to detect a weak x(t) signal in a particular 2 MHz frequency band: [f  1MHz, f + 1MHz]. You have access to a Nyquistsampled version of y(t) = a x(t) + n(t) where n(t) is white gaussian noise of unit intensity and a is much smaller than 1. Your challenge is to give an algorithm to decide whether x(t) is present with a false alarm probability of at most 1/1000 and a probability of missed detection of at most 1/10. * Assume you know what the value for a is and x(t) = sin(2*Pi*(f+100kHz)*t). How many samples do you need to look at (as a function of a) in order to meet the target reliability? How would you do the computations? * Now assume that you do not know x(t) exactly. Rather you know that x(t) = sin(w t) where all you know about w is that it is within +/20kHz of f+100kHz. Approximately bound how many samples you now require as a function of a. How would you do the computations if your goal was to minimize the number of samples required? * Repeat the previous part, but now your goal is to minimize the number of computations required. * Suppose that you had access to a processor that could do a fixed number of MFLOPs. As a function of a, how would your strategy vary if your goal was to minimize the amount of realtime required to get the answer? (count both time to sample and time to compute) While working out the above problems, you are allowed to make whatever assumptions you feel are necessary that preserve the essential character of the problem. 
August 6th, 2009, 11:20 AM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Challenge problems
That first problem is really quite interesting to me. Quote:
 
August 7th, 2009, 10:46 AM  #3 
Senior Member Joined: Dec 2008 Posts: 206 Thanks: 0  Re: Challenge problems
I do get your explanation . As the prices are uniformly distributed between [5 10] if we consider the prices are integers then it is becomes a discrete case and we can calculate the probability . I am actually more interested in setting up the model. Also i dont understand the final statement of the problem "Assume that if at any time the balance of payments becomes skewed by more than M times the maximum price of a dish in one direction, then unconscious feelings of being cheated will manifest and the friendship will grow distant." Wont C order for a costlier dish when it's D's turn to pay the bill so that their friendship remain intact.

August 7th, 2009, 06:02 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Challenge problems Quote:
You can calculate the probability regardless of whether the prices are integers or not. Quote:
 

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