W3600_Midterm_Sol

# W3600_Midterm_Sol - SIEO 3600 (IEOR Majors) Midterm...

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Unformatted text preview: SIEO 3600 (IEOR Majors) Midterm Introduction to Probability and Statistics March 24, 2010 Page 1 of 9 Midterm Place all answers on the question sheet provided. This midterm is open textbook (Ross) and open notes/homework. You are allowed to use a calculator, but not a computer. Write all answers clearly and in complete sentences. All answers should be supported by analysis or an argument. This midterm has a total of 100 points. First Name: Last Name: Honor Code I resolve i ) not to give or receive aid during this examination, and ii ) to take an active part in seeing that other students uphold this Honor Code. Signature: 2 SIEO 3600, Midterm 1 2 3 4 Total SIEO 3600, Midterm 3 1. ( Probability in Food ) (30 points) A careless chef randomly cut a foot-long submarine sandwich into three pieces at point X and Y , with joint PDF f X,Y ( x,y ) = c if 0 ≤ x ≤ y ≤ 1; 0 otherwise . (a) Determine constant c . (5 points) Solution: 1 = Z 1 Z y cdxdy = c Z 1 y dy = c 1 2 , which implies that c = 2. (b) What is the marginal PDF of X , i.e. the length of first piece of the sandwitch? Use the marginal PDF to compute E ( X ). (10 points) Solution: f X ( x ) = Z 1 x f X,Y ( x,y ) dy = 2 Z 1 x 1 dy = 2(1- x ) , for 0 < x < 1 . Therefore, E ( X ) = Z 1 f X ( x ) dx = Z 1 x 2(1- x ) dx = x 2- 2 3 x 3 1 = 1 3 . 4 SIEO 3600, Midterm (c) If this chef asks you to choose one of the three pieces before he actually does the cutting, which one would you pick to maximize the expected proportion? (15 points) Solution: f Y ( y ) = Z y f X,Y ( x,y ) dy = 2 Z y 1 dx = 2 y, for 0 < y < 1 , E ( Y ) = Z 1 f Y ( y ) dy = Z 1 y 2 y dy = 2 3 . Therefore, we shall compare the expected lengths of the three pieces, i.e., E ( X ), E ( Y- X ) and E (1- Y ). We have E ( Y- X ) = E ( Y )- E ( X ) = 1 3 = 1- E ( Y ) = E (1- Y ) . Since they are the same, we can choose any piece. (Maybe the middle part has more meat and greese?) SIEO 3600, Midterm 5 2. ( Probability in Sports ) (42 points) The annual Orlando Disney marathon takes place every January. (I participated and finished this year’s run on January 10, great experience!! http://espnwwos.disney.go.com/events/wdw-marathon/ ) During the run, runners run through four Disney’s theme parks: Epcot (starting at Mile ), Magic Kingdom (at Mile 8 ), Animal Kingdom (at Mile 18 ), MGM Studio (finishing at Mile 26 ). (visiting 4 parks in one day without tickets!!!) Some Disney characters (possiblly multiple for the same characters) are distributed in these...
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## This note was uploaded on 10/03/2011 for the course SIEO W3600 taught by Professor Yunanliu during the Spring '10 term at Columbia.

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W3600_Midterm_Sol - SIEO 3600 (IEOR Majors) Midterm...

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