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Unformatted text preview: 332:226 Probability and Stochastic Processes Examination 1 March 12, 2009 SOLUTION VERSION 1 You have 110 minutes to answer the following four questions in the notebooks provided. This is a closed book exam; neither notes nor calculators are permitted . Make sure that you have included your name, Rutgers netid and signature in each book used (5 points). Leave your exam paper stapled to your exam blue book (5 points). Read each question carefully. All statements must be justified. Computations should be simplified as much as possible. 1. 50 points SHORT ANSWERS: You must explain your answers to these unrelated questions. (a) A Starburst candy package contains 10 individual candy pieces. Each piece is equally likely to be red, orange, yellow or pink, independent of all other pieces. What is the probability that a Starburst package has only red or pink pieces and zero orange or yellow pieces? Each piece is red or pink with probability p = 1 / 2 . The probability of only red or pink pieces is p 10 . (b) For events A and B , P [ A B ] = P [ A ] + P [ B ]. What is P [ A | B ]? In general, P [ A B ] = P [ A ]+ P [ B ]- P [ AB ] . Hence P [ AB ] = 0 and thus P [ A | B ] = P [ AB ] /P [ B ] = . (c) K is a geometric ( p = 1 / 2) random variable. What is P [ K E [ K ]]? K has PMF P K ( k ) = ( (1- p ) k- 1 p k = 1 , 2 ,... otherwise = ( (1 / 2) k k = 1 , 2 ,... otherwise Recalling that E [ K ] = 1 /p = 2 , we have P [ K E [ K ]] = P [ K 2] = P K (1) + P K (2) = 1 / 2 + 1 / 4 = 3 / 4 . (d) The temperature T in this thermostatically-controlled lecture hall is a Gaussian random variable with expected value = 68 degrees Fahrenheit. In addition P [ T < 66] = Q (1) = 1- (1). What is the variance of T ? P [ T < 66] = P T- 68 T < 66- 68 T = - 2 T = Q 2 T = Q (1) . Thus T = 2 and T has variance Var[ T ] = 2 T = 4 . (e) MULTIPLE CHOICE: For the course textbook (Yates & Goodman 2nd edition) did you (A) buy a new copy from a bookstore or online bookseller, (B) buy a used copy, (C) beg, borrow or steal a copy, (D) survive without a copy, (E) do something else (if so, then what?). Except for answer (E), you need not give any explanation. All students providing an answer will receive full credit for this question. By the way, its perfectly OK and completely logical to save money by buying, borrowing or begging (but not stealing) a used copy. Im hoping you didnt answer (A). For each new copy sold, I have to donate my royalty back to Rutgers accoding to a new NJ state ethics code. Im looking forward to reading any explanations for answer (E). 2. 40 points In a youth basketball league, a player is fouled in the act of shooting a layup. There is a probability q = 0 . 2 that the layup is good, scoring 2 points. If the layup is good, the player is also awarded 1 free throw, giving the player a chance at a three point play. If the layup is missed, then (because of the foul) the player is still awarded one point automatically and is also awarded one free...
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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