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Final_2009_Solutionsx - Theory of Probability Prof.Dr.M...

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Unformatted text preview: Theory of Probability Prof.Dr.M. Nahit Serarslan Res. Assist, H. Kutay Ting, Res. Assist. Evsen Korkmaz Res. Assist. Pinar Dursun. Res. Assist. Ronay Ak 25.05.2009 Final Exam Duration: 150 rnin - Please show all your work and use verbal explanations, if necessary. 1. Suppose that before observing new evidence the hypothesis H is three times more likely to be true as is the Hypothesis 6. lfthe new evidence is twice as likely when G is true thanit is when H is 10 pts true, which hypothesis is more likely after the evidence has been observed? PM): 3/98) (net Swine 9;: a.) Praia): 2. P(ElH) (gown where.) P[ P/H/E) : fl _ PKHE) _ PUMP/EM) P/QIE) Wee) PréE) _ r76). P/Elé‘) Hence , Al ,3 more L249 afler‘ +Ae €Vr’ofizOCz has been 05W. Final — Page 1 of a certain type is Poisson random be observed if it is the leaf. if we let Y an insect gs laid on a tree leaf by h a random variable pan only 2, The number of eg r, suc iable with parameter A. Howeve var 0 then we cann positive, since if it is e the observed number of eggs, P{Y=i)=P{X=ilX>0} denot then w . . . here X IS Paisson wuth parameter/l. Find E[V] Pfyz [I X>o§ = ?iX:L',X>0i P{X>o§ EM= :— mas} 4” P§Xyoj = Efxl five? A —' _——-—.—- 1— gr? Final # Page 2 3. The following dartboard is a square whose sides are length 6. The three circles are all centered at the center of the board and are of radii 1, 2 and 3. Darts landing within the circle of radius 1 score 30 points, those landing outside this circle but within the circle of radius 2 are worth 20 points, and those landing outside of circle radius 2 but within the circle of radius 3 are worth 10 points. Darts that do not land within the circle of radius 3 do not score any points Assuming that each dart that you throw will, independently of what occured in your previous throws, land on a point uniformly distributed in the square, find the probabilities of the following events. a) You score 20 on a throw of the dart. b) You score at least 20 on a throw of the dart. 12 pts c) You score 0 on a throw of the dart. d) The expected value on your score on a throw of a dart e) Both of your first two throws score at least 10. f) Your total score after two throws is 30. N Area (s.)=36»9r (0 M) Lei Xc'deno-ié your scan: on ,‘Jh +6rou) [4,: 1,1,...) 0 X: :31’ -1: ' -;3. . 40.21 )PE .20} 36— ,4 0/) thJ 03%;,20335», 36 34 ’ '3— 9 . y a) P5X.-=o} = 35;.”- , ,_ 1: a) Piwg2rols/I—F{)¢.o}),(4-figxfo} a q 2. - O _ Yrs-Fl) 4—291 F) flaw/”530} - 2.(P§X-=30,x4.+1:o}+ M120“ “4) 5/95/9320}; 3r+F - 7r = 357:- If] =2. l/7‘§—)+L.§I 36 49. 26 5. If 2 is a standard normal random variable, what is COV (Z, 22)? ' ' 10 pts Cov my) : E. [(X-Ech)-(v- 500)] COV (93,12): E[(2-EC23)- (%‘—EC%‘3)] Near) 049/ Mob/,0, 0,5 mag/ad normal random VCm'ob/e at 0 000/ f M { , Way. Vern?) = ECZ‘J {£1301 4 = E5217 ~ 0 Eff] :1 =>Cov (2-,?2)= E[% (22.4)] _ N 0 : tCa‘j l. 7— [5&3]: 23._L__¢'2%o/2— ,90 V517 1489’ij mh‘jm‘hon by [3015, E‘Jj usiry) _ 1.7: flea 2 e zen/2'0” . Finalm Page 5 6. The joint density ofXand Y is given by -2x d-Sy raw—‘65 Cl<x<eoo<y<oo 10 t5 0 of 'zeruflse p Find the density function of the random variable X/Y. F(>(/\/)= P{X/‘/ gag Off 5 6.29% 33mg =]( 361 ~2’Xe SJCB d3 Final — Page 6 7. The neighborhood bank has three tellers, two of them fast, one slow The time to assist a customer is exponentially distributed with parameter r). = 5 at the fast tellers, and = -l~ at the slow teller. Jane enters the bank and chooses a teller at random, each one with 10 pts probability 1/3. Find the PDF of the time it takes to assist Jane and the moment generating function 3-1:?) L51" X (21.20046 Sen/Ace #er / \/ demo—[c Adler; fix m = 2‘. _€,y{"/y)- WM} VJ Harm/74' jeflmb’y Anchbj of XI 00 44% -4 ‘ 04X Mbk): Efefyffaty/é. 66 ’73:?ch Final — Page 7 8, Your company must make a sealed bid for a construction project. if you succeed in winning the contract (by having the lowest bid), then you plan to pay another firm 100 thousand dollars to do the work. lf you believe that the maximum bid (in thousands of dollars) of the other participating companies can be modeled as being a value of a random variable that is 10 pts uniformly distributed on (70, 140), how much should you bid to maximize your expected profit? [F JO” 5’5! 70 yoéaré/l/O , Ham you W'” W,” +hc 5'6 0”!” Make a foroFH— 070 ar— #00 W’V’Lh imbue/"w {ma—«Mm ) Or my, +A¢ 5:2 we Page 5""?! (”who and more mm 0 MM «>70 0. HenC€ ,UOUF CXPCCJCDI [04974; v13, —4—— (a- {00).(4140- x) 3 3L0 (2mm - 93— 414000) _4._ 4O D’iL/‘é’mh’af—lty 000/ Sci—H29 1‘41"— [OfeCéeol'l’B 479 O ja'VES lhO—lx; O Winona / you should 590/ 0.0 'Ilhoujono/ c/o/lo/S: Your expgcko/ PM 72+ W” #120 A; 4/0/7— muse/m] elollarS. Final — Page 8 9. The joint density of X and Y is given by fol-I) = {xe‘o‘ffl x > 0,); >0 0 o {Fae-r351: a Are X and Yindependent? 4mg) ~ ) £/%)=fm”w = 0 O -— -7( é’Xe'Oo— [—X 65“) : Ke'k 9° —-{9r+) 4”” 79(7): fate J V 0/9?sz 0 ‘ oo 00 = - q @,{9<+y)/+fe_(«+fld G D 4 = _ -(r)(+y) Do a L —_':i = 6 10/003): 6/4171; 6/) - (1+3) .4! .3 “e Okete :W-e 30/ 10M Ya/E Ihmduqc~ Final ~ Page 9 10, A game is played using an urn that contains a number of blue and red balls. On each play of the game, a ball is chosen at random from the urn, and then replaced along with two more balls of the same color. You play three times, starting off with five balls of each color. Being careful to identify any conditional probabilities that arise in the course of your solution, find the probability ofthe following events: a). All the balls are red. 5) The last two balls are red. Lew" Xi cit/mic W eva' O? chained/13 a red ball on Me Hi, Way (as. /.2,3), 0) P2m3=fi;' - i,_}_.§__ A ’11., 11 10 t _3. 46 ’9) P§X3XLX4E= FEXglxzxf}. Pflexf} PM} — 3; i .‘E. , .5. 10 pts L Final — Page 10 a ...
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