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Quiz 6 - yr 1c l ’'1 Al 2602Q6.nb a 1 Math 2602 Math 2602...

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Unformatted text preview: yr, : 1c . l ’ '1 Al 2602Q6.nb a? 1 Math 2602 Math 2602 fiaril 18 ame: ZQgEP bE \_/ Section —-Cirole One: Jose Miguel Renom Marcus Sammer S. Singh 1 J. Dickerson Safabakhsh Open Book and Notes. You have 50 minutes. Carefully explain your proceedures and answers ' I Open Book and Notes. You have 50 minutes. Carefully explain your proceedures and answers All answers shouldbe given many (no decimals) if possible. All Logs are base 2 . I Problem 1 (5 points) 3 2 1 Jobnlnzcrwould like you to find the LU decomposition of ( —3 —3 —3] 9 4 1 3 z 1 r2443” 3 2. 1 (BAZr'Z. 3 z \ -3 «3 -3 ~> -| «I --2. D a; ..)' .52. . L ‘ 0 O, 3 a J IAnswe 1'- r -l l O U:- o ..-1 .. ' Math2602 . Apn'l18 3 2. ‘ o O 2 2602Q6.nb ~ r . v; we Name: , E Section --Circ'le One: Jose Miguel Henom Marcus Sammer S. Singh J. Dickerson Safabakhsh Open Book and Notes. You have 50 minutes. Carefully explain your proCeedures and answers ' I Problem 2 (10 Points) Ricky Dale Crain would like you to write down as a markov chain: Suppose we have a car rental company with offices in LA, NY and Atlanta. Cars are rented out on Monday, and returned by the folling Monday. The probabilitiy that a car rented in LA is returned to LA is 114, and returned to Atlanta or NY is 3/8 (each). A car rented in NY is returned to NY with a probability of 1/3, and is returned to LA or Atlanta eiach'with a probabil- ) ity of 1/3. A car rented in Atlanta, is returned to Atlanta with probability 2/3, and'is warmed to LA or NY each with a - prabability of 116. Write out the matrix of the markov chain. Find the long term probabilities of the locations of carsJust write out the euations you have to solve. Don't actually solve them I. N ,4 Azl . _ L 32.5. l/3 I/é, «75' 'l W L A; "I 3’9 V3 '/6 Aer: 3 _, A 3/9 3/3 -2/3 I/é, A, .-0 [/3 , 3/8 [/3- A A _L I . —,1—§L+3N+Z,4;Q ‘ 3 2.. 1 ,. v ~L———N1———r4—0 aeocehese t 3 e _ 2 :nd w: W55 .2- ..J- ______L __ be 0595’ g L 4' 3,_I\/ 3 ,4 -0 L+ N +4.: (- . 2602Q6.nb 4 ‘ 7- 4. ' . a | {I} Math 2602 April 18 gr Name: page [55y Section —-Circle One: - Jose Miguel Renom Marcus Sammer S. Singh J. Dickerson Safabakhsh Open Book and Notes. You have 50 minutes. Carefully explain your . .proceedures and answers I Problem 3 (10 points) Paul Anderson woule like you to consider the matkov chain: .. . '/.s‘ 1/5‘ 63 O a _ o e “j, (/4 a '/ 2. ’/z ' a 3 The above picture represents states in a markov chain. The transitions are: At 2 and 3 and the probability is 1/5 moving to the left one step. At 2 and 3 the probability of moving to the right one step is 1/2. The probability of staying at 2 and 3 is whatever it has to be, given the previ— ous. At the end points (1 and 4), the probability is 1 staying there. Write out thematrx of the matkov chain, with absorbing states written first. Find R and Q. Find the matrix which gives you the probabilities of ending up in the various absorbing states, given that you started in the various non—absoblingstates. Write it out as a products and inverses of matrices. . What entriy in this product tells you the probability of ending in 1, if you started at 3. Don’t actually find the product. ‘4 7— o 7- fl ‘ I _ 2. 3 ,3: n ( I: ~43) : - c Q 2 0 a 5 “'0 5‘ r 7 2' ' 3 I . - 3 3] The e“t'3 {4' the. ‘Fr‘t'ie r0“ 3 Second colVM“ will tell 30¢ eke Frolo OFenola‘ds PA I ‘Frbm 3 ...
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