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2008 fall final - Date January 9 2008 Time 15:30-17:30...

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Unformatted text preview: Date: January 9, 2008 Time: 15:30-17:30 . // Instructor: Dilek Giivenc MATH 230 F IMPORTANT 1 Check that there are 5 questi ns in y 2 Do NOT use your mob le p me as a alcu tor. T 3 Show all your work. Co cc results w t ut su cient explanation and correct notation might not get full credit. ‘ 4 Write your name on each 1 2 4 5 TOTAL é l 20 20 20 20 100 GO D LUCK & HAVE A NICE BREAK! 1. Jobs arriving to a computer server have been found to require CPU time that can be modeled by an exponential distribution with mean 140 ms. The CPU scheduling discipline is quantum-oriented so that a job not completing within a quantum of 100 ms will be routed back to tail of "the queue of waiting jobs. 3) Find the probability that an arriving job is forced to wait for a second quantum. (7 points) b) What is the probability that the third arriving job of a day will be the first one that is forced to wait for a second quantum? (7 points) c) Of the 800 jobs coming in during a day, how many are expected to finish within the first quantum? (6 points) , J, ‘ =4. #743 X> o a) A, Ago /(x) MD a X m - M ___/_0_ 00 “.25. “7?; I = 2 ’70:: e /7~ 0/4535 A90 : .Le/éOchs—e a “ F(X> ) [90,40 . ’ ° L) Yfi" 0/ cuff/WW d’obS ’10,ch 7Z0 AMHLL (”h/0M 1/40/71 I} 6/62”le 20‘ Wit/a, «9‘33: 7"“”"“”“ ~49~ \ ’f -éQJZ/Q \A/ MéL/fl'fisr 5Q ’9) flCy=3>= (/-—e 2 J / g“ f 405/05) may”) 0) V1.7? 0/ a/f/bfl/CL)? “[045 iii/[LEA W’Ml}! Magi/5?! 7VMVZVM /"/L gag 1’05:- , . ~13. a ng,a[/2:yao) p: be ’7 ) 5m): 3000' £453“)? 2 5700 (515/05) 7:. 403,35 Nr/ogJ/ggj' ’ e arerava NAME: .......................... 2. Suppose that cumulative distribution fiinction of X is given by 0- /\ >< /\ H ‘ F(x) = r—d |/\ HNIHWIMQ >< IV >< [\J N /\ a) Find the probability density/distribution function of X . (8 points) b) Let X , ,X 2 ,- - -X100 be a random sample of random variable X whose .c.d.f given above (or p.d.f given in part (a) ). Find the probability that sample mean of this sample will be at least 1. (12 points) ) la!) /.9/* (it/(41 /£K): Clix) 5% a l . , =F(l)—-LJWF0<) , (/ar x:1 pom) Hf : / — / 4:1” 3 Ma r 3—.— 3 6 .L 2— x /J/ 14x41 /C«x): dz“) = 3/: X (Q71 X51. -/*AV€' L‘s flékwrlfi'zwi‘ly faszt> (L 0‘4)“; .L L 3 —— a“): 19 x4 gm [144“ 24‘ g/+(/,L)+ZL_ L 12. [4x42 0 I 0 Oily/$911245; b) 500‘ o/Lclx-f-Lé fl+ IfLelx.—.f+ 50”) DI/L/ ;dK+/"g’~+f X24x=g+i+fij~ f?) ,8 M Va/(XD: 1144— //.0{)7'= /,44_/, M1- 0,2,1 . 1’ 0,2 _ 2, r.:o,osz =3 “XL-Log , rz_ 7-_o,0c9 7—, x /90 .7 ._ b1 C'L ,/,0g . ,_/.S /(2<>i)wfl(2; 30$; 2/42; 0 I : /, £00737: 0.9?82 NAME: .......................... 3. Let X l , X 2 ...X,; be a random sample of random variableX , Where f (x) is gamma density with parameters a = 3 and unknown [3. a) Find the maximum likelihood estimator of [3. (8 points) b) Is the estimator in part (a) an unbiased estimator for [3? Why? (6 points) c) Find the variance of the estimator in part (a). ( 6 points) *3: 'L ”.5. 00: 3X26 fl: X38 5 )4>o 6/ Ff3)fl \ , 2/3 x: 2 _X’ a —,— 77 X/ Z /3 MM ’l" 2’63 ,, M fig“) —— fiat/Le Kflfi’): énz), (/5), [2; {2/33 X , Zagfl/cfi-az—Bfifl/awfl , ’7 Kay. win/9‘ a; 7.22 #XL—Ifl '— fl 0/144 )- ,§L+E_>%L :0 Z u—‘Bfl/fi 4/3 /3 /\ zfi/ _ 31‘? m E}? w 2 3333 at: 355' NAME: .................. 4. In a study of estimating the average response time of a web server, 16 independent eXperiments are conducted and from each average of successive response times are calculated. The average of the results of 16 experiments was found to be 0.68 second and standard deviation was found to be 0.05 second. a) Estimate the true mean response time of this web server with a 95% confidence interval. (9 points) b) Estimate the true standard deviation of response times of this server with a 90% confidence interval. (8 points) c) State your assumption(s) for your answers in parts (a) and (b) to be valid. (3 points) .015 ___. 9 65 + flfl27— 0,58/77: {.2 /3) ”25 0/ w (max/9,320?) 2 {5", , 5) am 509% > 29,9; 2' 2%. x05 90 5/6. C.f /9r y— ! 7;” (W0 2%“ > ”MVm > M ( 0,0357%) 0,072) 0> f}; flwfo‘é a) wA. use. {'— q/Ks/ A; M 1 i :13 ' 71/ » 041a.” Ila/m may WV.” 3 t C yfl'xk’fl [Aavll' /;25 9f! 32— W; I 3 61¢ C / WA L 19 / /. A q r) V‘flfi/ flJ/I’VI :? “gt/Wk r1 Cl, ‘20 of ) of I U / A Oct/1 wflS/Y‘ucufi‘ (:"f' fa,” f“ /_3', IILAL /fi&/¢J W ”fin Vt/z/ NAME: .................... 5. A program’s average working-set size was known to be 50 pages with a variance of 900. A reorganization of the program’s address space was suspected to have improved its locality and hence decreased its average working-set size. In order to judge locality—improvement procedure, 100 samples of the "‘improved” version of the program’s working-set size were taken and sample average was found to be 45 pages. ' a) Is there enough evidence to believe that the reorganization indeed improved program locality? Test at or =0.05. Calculate P-value of the test. (10 points) b) Calculate the probability of making type 11 error in testing the hypothesis in part (a) when uA = 47. (10 points) #:1ch )7?%5 wiawo) F230 0» awe 2,1: ”Jab/fl # 1' [(4450 'c‘ 30 /r.'____“ \z/oa §4haa 2M: =»// 572 4 ,2“; /% MflVb/Qé W /€0(‘ adv/'M'lgn 74””f/0Ufi-J fifty/M ) flaw/(4%- fll/alVL: fl(24-/,Asz);—_ <9,fl¢;5<405’ ,1/0 2011 filo hip/«J AWL A) /(7&fz,,[£}mf)=p(09 ’L/7 I > ,— WAM jfi4~w¢5 We /7'em} 1% 30 [00 f 27450, (/I645)(3)=¢5,:955 L9 K f: ”6;;éj; fl()7>45,05{m’%?):fl(3> 3% ...
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