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Quiz 4-5-6 Solutions

# Quiz 4-5-6 Solutions - Bogazici University Department of...

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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 Name: . Surname: N E For ‘03 5.9.3:». 4 (Khxqj 1£C\$Og€o~d mus‘)!‘ H5. 00 «V @x‘uo ; fgcmam = j 2.6“7- = 2”“1 °f“f* I .00 x4 0 K 06 ‘1— ’CuC“D= fﬁCvnﬁth‘H afzou‘ —; 1:2 ¥1<¥2<l -00 .9 0.0“- .54 Pa 9‘ 4| ‘ﬂq, { . ‘DC iqw.q'€) 3‘— ]; 2d*16*1 c/fL'ﬁzA‘i‘l =)\-(E-)x‘:@ 036 ° ‘9 s - 7 2;" 5‘4’ #10? dag = £03 *1) A71: @) A945 oru. ., W S W“- as 32): mama ’3 043 MW» (7‘) .— eom‘ oﬁx ”C? (53‘Ra :3; 7; (y. D)=(;( (*1€'(3 ¥ ’ o 0",. a (-Sf}! Fwd C“3 {’90 i 7‘ .91 é 0% 9' )rféﬁe-JEJQAvL J ’SAD é¥&* ‘ 05 I“ XNLQ v—l—ps 9+ S 05W 'W /- W ® Fm “KW? SD‘é‘MX 00’} oazh’ i 00 _ ’ — (+ 00 ‘ W "533an = 5 [#33845 =[4 3*51’1"), ”3 = '41::(2 1? " 014.0” 59(1) " ’7 ““1009" a -: If} wave] 633% '2.“ Q“ (u),£“+aéa’z4=E O "2' Bogazici university Departmth of Industrial Engineering ' Fall 07. Probabilityfor 115.13 255 QUIZG __ K E \l " ,— Name: Surname: Swimwhhastickoflenglh Lwebmakitatapohtvdxidnisdnunmdmﬂyuduniﬁmﬂymﬁlmw bepﬁepiechatcontaimdncleﬁmdofﬂn sdchWe ﬁlmmpeatﬂnempmondnﬁckduwembﬁ widuWhaismecxpededlaudxodee stickdnatmmleﬁm’hakrbleakimtwice? (Hinclct chthelmgthoflhe stickaﬁerwebleakfonhe imtimc. htXbelhe MMthcnomdimSo dneqwnionisﬁxdingElXLYm maytakeximofﬁxd'ngandusthleYlhymrm-t) (No partial credits wimout ﬁnding the answer analylically withdle ridxt usage of mathanmalnomd) Compute ElX'IY-yl ifthejcintdmsityofx aninssivenby Mﬁ-ewequ 0<x<°°,0<y<°° - 1 (Hinf: At some point you may obscure that someth'ngis dimibuted with paramem l/y) Ifdne distrbution of X is given as the bllowing 3 rm= 5 1 ; lfxs-I Ifxso (Ix-=1 Fall 07, Probability for IE, IE 255 Quiz 4 Bogazici University Department of Industrial Engineering Kg «7 _’ f.— 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 e-mail 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 e-mail 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 e-mailed 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 ‘@’CM (‘pbuﬂgn‘i‘j Iﬂ‘k'eAs;&() .74 OCLQfﬁnLL 08S (1.9" OW ( e Mata-r; Folk Fn0*}°r\ i“ Cong-koa‘i‘ ”Wk km. ﬁe . A\ s‘lrﬁ‘av’hm‘ ‘ 7t} FC‘H A‘a (4)9 £00 Q’rPOM*ia\ / 7\ Mil-MW; “Wk fwd)”; 11w: 9 4444") %(Q>\ a ﬂat (/(6 (quﬂ‘), 4(gzrucu/wx') 4/ 7Lc: 4/20 , ‘i 0‘ 5 45, (0Q) E(A\;49:l’7"* ’7 M, h IR: in at: s/w (DOA "‘- 4 a: ‘_ pLg>c>®+P Ma “9“ L6] @mgﬂsweﬁt A» a,“ 009 «- 9013)!)- PLDM )Lc—v 3 31“” L1) '5 3,. /‘” s’lBeC ‘Wﬂﬁ aha \Oﬂ IS~~ k MM 5.45905. (C((N: M‘hﬂa‘) :Kuck “has Firm a CIA " .5) ' i 9CX= 20)’~‘—’— P(i5,‘345<<7-°- \$9 ‘Pr (1%“? C mad WW acquit. (“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, i” 6 ...
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