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Unformatted text preview: NAME: NetID: Signature: 332:226 Probability and Stochastic Processes Final Examination May 8, 2009 MAKE SURE TO READ BOTH SIDES OF THIS EXAM PAPER. You have 180 minutes to answer the following questions in the notebooks provided. You are permitted to use one double sided handwritten sheet of notes but no other resources. Make sure that you have included your name, RU NetID and signature at the top of this exam paper (5 points) and on the cover of each blue book used (5 points). Read each question carefully. All statements must be justified. Computations should be simplified as much as possible. Gaussian probabilities should be expressed in terms of the 8( ) or Q ( ) functions. When you finish, please submit your blue book(s), with this exam paper inside. 1. 40 points ANSWERS IN THE BOX: Each of these questions must be answered by filling in the corresponding boxes below. No justification is needed. No credit will be given for answers written in the blue book. (a) You flip a fair coin 4 times. Each flip is heads h or tails t . What is the probability of the sequence htth ? Since the coin is fair, all 16 length four sequences are equiprobable. So P [ htth ] = 1 / 16 . (b) X is a continuous uniform ( , 5 ) random variable. What is P [ X < 2]? Since X has PDF f X ( x ) = 1 / 5 for x 5 , P [ X < 2] = Z 2 f X ( x ) dx = 2 / 5 . (c) K is a Poisson random variable with E [ K ] = 10. What is P [ K = E [ K ]] Since K is Poisson ( = E [ K ] = 10 ) , for k = , 1 , 2 ,... , P K ( k ) = k e- / k. Thus P [ K = E [ K ]] = P K ( 10 ) = 10 10 e- 10 / 10 . (d) Y is a Gaussian ( 10 , 1 ) random variable. What is P [ Y = E [ Y ]] Since Y is a continuous random variable, the probability Y takes on a specific value is zero. That is, P [ Y = E [ Y ]] = P [ Y = 10] = . 1 / 16 2 / 5 10 10 e- 10 / 10 (a) (b) (c) (d) 1 2. 40 points In the Monty Hall game, a new car is (equiprobably) hidden behind one of three closed doors while a goat is hidden behind each of the other two doors. The game proceeds as follows: You randomly select a door. The host, Monty Hall (who knows where the car is hidden), opens one of the two doors you didnt select to reveal a goat. Monty then asks you if you would like to switch your selection to the other unopened door. After you make your choice (either staying with your original door, or switching doors), Monty reveals the prize behind your chosen door. To maximize your probability P [ C ] of winning the car, is switching to the other door either (a) a good idea, (b) a bad idea or (c) it makes no difference? As this is a popular question in probability and you may already know the correct answer, the grading will be based mostly on the quality of your explanation. To explain your answer, suppose the door are numbers 1, 2 and 3. Let H i denote the event that the car is hidden behind door i . Also, lets assume you first choose door 1....
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This note was uploaded on 02/13/2011 for the course 332 226 taught by Professor Staff during the Spring '08 term at Rutgers.
- Spring '08