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Unformatted text preview: Bogazici University Department of Industrial Engineering 1) The joint probability density function of (X1, X2 ) is given as: ' ﬁx], x;) 3?. 0s X15 xzsl 0 o.w. a) Are x1 and x2 independent? (4 pt)
b) Find P( x; > 0.75 ) (3 pt)
6) Find E[X;] (3 Pt) < 2) LetXbX1,...,Kummdmtegmﬁdrandonivmiableshavingaeommon
parameterl. Determinethedism‘huion ofmin (XI. X2....,X.).
Hint: 'I'heprevious PS? (4n) , 3) IfXandeindependemaqionenﬁdrmdomvmiablesepchwithawmmonparamemr
A=1,computethejointdensityof a. U=X+Y(3pt) b. V=X/Y(3pt) Fa1107,ProbabilityforIE,IE255QuizS K \( N
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(Ix=1 Fall 07, Probability for IE, IE 255 Quiz 4 Bogazici University Department of Industrial Engineering Kg «7
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Name: Surname: 1) Aﬁer your graduation from the IE department, you become a management system 40/? specialist in a service company. In your ﬁrst day at work, your boss sends you an email
in which he says that they have a problem in the Tokyo ofﬁce, that they were not able to
calculate the probability that customer C is served second in the setting described as the
following. There are two identical robots (servers) in the oﬁice and there are three types
of customers served in this facility type A, B, and C. One type C customer comes and
finds that two robots are busy one with type A and the other with type B. Aﬁer
understanding the question you skimmed the wordy email and found invaluable
information: They company observed that on the average it takes 10, 15, and 20 minutes to serve a customer of type A, B, and C respectively. And further they observed that in the
service of customers, the service time of the customers does not change with the
knowledge of how long it took to serve them to that time. an a. Having taken IE 255 and attended all of the classes and PS’s, you emailed your
boss, with no hesitation, that the service time of the customers seems to be
distributed with what? 1,} b. You support your argument with knowledge of the hazard rate function’s some
property. What is this property?
c. What is the hazard rate function for type B?
Z: d. What are the parameters of the distribution for A, B, and C?
ﬁn. What is the answer for the question that customer C finishes the service second? (7’[email protected] €*€:>ne_n¥‘~a\ b‘ﬁﬁbu‘lﬁon (1
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(“50:07.55 .. 9(‘4'4 '5’2’4 a". ‘ — 10,33) 'EG‘K‘”) : 2. i‘idﬂﬁ) —L = 210.3(04 = 55— 41°) pmbabilitythatxisbetmen1/4and1/2whena=b=1 .. is,asseenintheclassnotes,givenasthcfollowing. Calculatethe ”I" 1° m,
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 Fall '07
 AybekKorugan

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