2008 spring final

# 2008 spring final - eEsEﬂ/E Copy 2 Date Time 17:30-19:30...

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Unformatted text preview: ; eEsEﬂ/E Copy 2 Date: July 25, 2008 Time: 17:30-19:30 Instructor: Dilek Gﬁvenc /‘ f IMPORTANT 1 Check that there are 5 questions in yo boo 2 Do NOT use your Imobil’é phone as a c on tor. T ' it off during the exam. 3 Show all your work; Correct results wi out suff rent explanatiorr and correct notation might not get full credit. / 4 Write our name on - 1. A box conﬁgured and m white balls, Where 11 , m > 0. a) \$650 balls are randomly drawn, What is the probability that they are the same color? (6 points) b) If a ball is randomly drawn and then replaced before the second one is drawn. What is the probability that these two balls are the same color? (7 points) c) Show that the probability in part (a) is always smaller than the one in Part (b). (7 Points) n I, ' m I V. {91 : T43 3‘ a a) h m‘ T fgibnq\ {mam}! E ) (‘7’ ) {ﬂare-2W1! nemlc’n-Lm—z) A ‘0‘ mﬁﬁ/ﬁf \ C H; 9f > H0 X! ) ’ _.J_;L-_’ ” r. 12”? b) / _ /n+m)/n.;.n7> (/7+y;q\ z \ 1' ' 'f Mir (6/. 01C?! {:5} I; a)" ﬂax/war Cﬁfommmr‘u/ //r ‘p - / J n ‘\ “WWWél-Féﬁ" 0% (I?) 1": Kﬂ+m—K)[ﬂ?¥—m 1) 1 3 2 1/7241”? +m/7l—i—m WE, wM/ﬁaxé. @ "3mg! [email protected] @r @_/7ﬂ7__mn :2) ﬂxoé 5‘1 CLEJWWS‘ S‘mmaaizef. (maﬁa, 2924.35) @ U NAME: ........................ .. 2. There are two types of batteries in a bin. Lifetime of type i batteries has Weibull Distribution with parameters OL = 2 and Bi, ( i = 1,2 ) A battery that is randomly chosen frorn the bin will be type i battery with probability pg, 2 then 2 pt. =1 . If a randomly chosen battery is still operating after t hours of '=1 use, What is the probability it will still be operating after an additional 5 hours? r Ck: f) .i /9(X:>:~‘f'§ ) X/L/ (01:2; [’35) I 5' ﬂgl 4/25 r 0‘0 342,-! _2XLIq/ _.2.-a I'd/ Z F(_X>él£’3?¢é):/2/gfg§ .33 X r a g ) ..— 2(5ff.);€ﬁ I 5 £— mu' :éZ :/,z. 51 W00 + l 0} 415% m Pmm ﬂ 5— 7L f?— a p‘(X>5TLLL My” : "Zr—’3; -233; la, ’7'“ 2.?— -2-é/g/ _2f/€Z :ﬂa +ﬂ€ NAME: ........................ .. 3. Let X be uniformly distributed random variable over the interval (0,2). a) Find the failure rate function of X. (8 points) b) A random sample of 49 measurements was taken on random variable X. What is the probability that the mean of these 49 measurements is larger than 1.1 ? (12 points) (/rmzéu 04x42 ,1 ‘ A. “, 95+); KG“): F(X>:~ : ale/h a) J‘ gig). f —L(2-—f\' ._ 2 J :24 _ or... 0442 we M féKZ—ﬂ 2 if (3)2 4 _ / (7—2; 2’ "i W; Wvd/Z"? X Vi- : M“ )4 p?— 5> m ,1 "L “L / «t:,_/_—:a0525 .7: 3 Liv—#- 4/}; r2 [Vi—(H: W 3(45) r g b”? // r \ r 5 , ‘ﬁ _ r P a I}; /9(;?>/,/)m/(Z\ mm y/(J/ ) g / Haw) : /——C?.§’§63 — ty‘,//3/ NAME: ................ . . 4. Lag time in a printing queue (the difference between the time that a program is placed in the queue and the time at which printing begins) at a particular computer center is normally distributed random variable with mean p. and variance 62 . For 6 programs the following lag times (rounded to nearest Whole minute) are recorded: 1 7 20 1 1 14 8 21 a) What is the sample mean and median ? (5 points) b) Estimate the true the true mean lag time at this Computer center with a 90% conﬁdence interval. (10 points) ' — gg+20%/f%/4%E’+Zl :/5,/;Z [fl—- __. r- /l/’:eo’:/Faba : Emir- : /3 .3 2.- 1 “7‘ 8575.}; "ten.— /5' I?) /5//* _ _ a 5: ELK/LIL: Fri”; ) by 4 M:25.04532 xi 0 : /__{_,5 325/03: ﬂ:g(;W/;{/ d/ o I /§\=2 53/5 /—"0L: fzatat .1 m h u fl I: ﬁﬁf‘r’l-‘Ju 9O 0/0 C L ﬂ (ﬂea/w mar / —L g 3 /§f?;_/2.0/S‘ fglfii ) MM? / 4 i via/[6 NAME: .................. .. 5. The weekly downtime X (in hours) for a certain production line has Gamma distribution with parameters on=2 and [3. Let X; aXz ,. . .Xn be a random sample of weekly downtime ofn weeks. a) Find Manimum Likelihood Estimator, [5’ of ﬂ. (10 points) b) Find Va.r( ii ).(5 points) 0) Show that ﬁx? i=1 3n is an unbiased estimator of VarCX). (7 points) _ d) During six weeks the following downtimes were observed: 4.1, 3.7, 6.0, 6.9, 4.0, 5.8. Find the Maximum Likelihood estimate of ,6 .(3 points) / 2—( __.L a) 10"): 1X €— ﬂ X>o a Fog/5 , ’44-— jfz‘f‘: -' X3 /‘5 (/ ﬁg W n '1 ., ——_"Lz_ AXKKE}: 7/— X: 2 x3 x'; {57- ” ;£\\ x.’ {a I {Kr/a): aiaéxnsl: 45,4 we 7 n —..>£t'_ 1 _ 5' £1 (mg, .r ): imﬁxﬁﬁﬂ/k iii-:3 lgz {Lt}- : Z&X¢_2ﬂj4:P—E:, / mm 2 o_ 2.. 7. ZXLMO 0% e —2mg7ru2xzr:0 I w ,’*_ ZX,’ F 7 :D/Vléﬁﬁﬂiﬂwemrz ...
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2008 spring final - eEsEﬂ/E Copy 2 Date Time 17:30-19:30...

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