PracticeTest1Soltn - Practice Test Solutions 1. (36 points)...

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Unformatted text preview: Practice Test Solutions 1. (36 points) In a factory, the engineers described the time to failure of a machine (in hours) as a random variable X having a distribution with the following probability density function: 3x2 ft (x) = 124 ’ 0, otherwise 15x55 (a) (5 points) What is the mean of this random variable X? 5 S n "2. Li 3 EEQ— 3 i” i’l—Li . 44% 4% (b) (5 points) What is the variance of this random variable X? . m _. 1.. .u-' 1' I" -- -~ * X VNDQ‘ Liz‘l Lil gift]: BELM = i’lL‘ \imiifla ism, ~ Law" = oszsg 13M; (c) (7 points) What is the cumulative distribution function for this random variableX? ‘5 3 ‘3 '5 FL B £— AK : FL“ 1 2—— -' "L U5 1" nu I'LL: 1 l’Ll—i 11L! (d) (3 points) For this random variable, which is more likely: a value near 2 or a value near 2.5? Why? ((1.697 / S.) a \JCIKUE. Jose; A0 is more \lkdfi 0A1 Moi (e) (5 points) What is the probability that time to failure (X) will be more than 2 hours but less than 4 hours? A numerical answer is expected here. Plza MM :: Flu, Fm) 7— 5%: % Tu: fl 1% 1 er, rm (f) (8 points) Give an expression for the inverse CDF of this random variable. Also generate a value of X from the inverse CDF when a random number R is equal to 0.5. _. :2- 2‘:— FLle Fmskl 3 M! M (g) (6 points) Currently the range of this random variable X is [1,5]. If the range of X is [4,8], determine the mean and variance of new variable. (Hint: Use the results from (a) and (b) to find the new mean and variance.) E l: it‘ll :— {97} ‘8 VOrDWQ— \JOM‘E741: 0587,53 Id 2. Var(X)=1; Var(Y)=4; Corr(X,Y)=0.8; Var(2X-3Y)=? Vol“ (1% ‘fQii -: 4 “Jet L70; +05Varivii - li’ll ’5 Cowl LXI _> .2: 2,0 «Hg ,— 3. The following design is proposed for an automated manufacturing cell: A new job will arrive at the cell at precisely 30-min ute intervals, and jobs will be processed first-oome—first—served. The jobs can be classified into two types, depending upon the amount of work required to complete the job. Job Type 1 requires 20 minutes while Job Type 2 needs 50 minutes. The system doesn't yet exist, so it is not possible to obtain a sample path. However, it is believed that a reasonable model of arriving jobs is that they are equally likely to be of each type. Therefore. flipping a coin can generate job type inputs of this simulation. if the head of a coin appears, we assume that Job Type 1 arrives. Otherwise, we assume that Job Type 2 arrives. Simulate 5 jobs arriving at this manufacturing cell. (a) (10 points) The event of flipping a coin is similar to Bernoulli event with probability parameter, p= 0.5. Find the inverse CDF of this Bernoulli distribution. Then generate 5 random variates from this Bernoulli distribution when R = 0.2, 0.8, 0.1, 0.5, 0.9. (Hint: p(x) = 0.5 if x = 0 or Head and 0.5 ifx = 1 or Tail.) Le_\ x i 0 [0, MA) X {fix} Fix] v<\Ltn0 [F0] " l L \ I y <’ 0mm) OéQQU-S 8:01 03 at as as H Al" i5: 3 (b) (10 points) Using the sample path you got from (a), perform hand simulation of Sjobs (Note that if your generated variate from (a) is 1, it means Tail appears thus Job Type 2 arrives). If you are unable to answer (a), then use the following sample path. H,T,H,H.T Rrr‘wofi Thus: ’50 , LO , '30, [20, 160 i ‘MLs-.’LO,SO 10,20 ,90 Sam! cc, Tl H T ,‘H H T dock QUQM- Lctll') SH} P Most} Rrr'ml Next Defl- 0 a O O 30 00 so R O \ go 50 50 D O O 60 0‘0 so A o i 50 “0 lO 0 A t l HO 1 30 D D l no IKO no is l ‘ ‘60 ’30 i'SO D t O l is; tea \60 p30 0 l m 200 200 b D 0 [0 pg (0) (5 points) What is the average number ofjobs waiting in the queue? up . (42 points) The following design is proposed for an automated manufacturing cell: A new job will arrive at the cell at precisely 30-minute intervals, and jobs will be processed first-come-first-served. The jobs can be classified into six types. depending upon the amount of work required to complete thejob, as shown in the followin table: Time To Comlete minutes a. (10 points) The job-type inputs to a simulation of this system can be generated by rolling a fair and six-sided die. That is. Pr{JobType = j} =1 j=1,2,...,6. Derive the inverse—CDF that 6 3 generates the job-type inputs. b. (4 points) Generate 4 job-type inputs when R = 0.1. 0.5, 0.6. 0.9 using the inverse-CDF you got from part(a). . (24 points in total) Write the letter of the correct answer in the space provided: (a) [)7 In the linear congruentiai generator Xi+1 = (4Xi + 3) mod 5, if the seed is 1. then the second pseudorandom number generated is (A) 1; (B) 0.2; (C) 2; (D) 0.4 Xia- {anew 5 w. ‘1. .. \ ,_ -- X1: {VI-It’ll“) Mad-3 =- l :7 PM? a 0'1 (b) D To be useful, the period of a pseudorandom number generator should be very long so that (A) we can't guess the next pseudorandom number; (B) we don't end up reusing pseudorandom numbers; (C) we can have streams of pseudorandom numbers that are far apart; (D) B and C; (E) none of the above. (0) True or We use chi-squared test to check if PRN are independent. C‘IAI'x Squad." 5' "t C Elf: 3r" L{{;“n~rli‘ou\"\ai\s (Qu n up - Ago-d A \‘l (d) True 0 The covariance is a unit-free measure of dependency .(Tar tr “I‘d-UH?) between two random variables. Corf'e\o\—\or\ ts amt “Free (e) True oTwo random variables X and Y are independent if the product-moment correlation between X and Y is zero. NO; 0 corrE’Xalrkpn gives A34 )0 (k9,? . Short answer questions: a. (4 points) A pseudo random number generator is given by Xi+1 = (4951 X; + 24?) mod 28. (i) What is a maximum period? (ii) Can we achieve the maximum period with X0 = 2? .g ' ’M” ': :1 (at) MTZ‘O owl Cit-‘0, so inflati‘tML/M (Hound ‘9 M E..— (a) 0: mm as or an aim a: nmiM—i— .3. SO (1A3,- lg U“ (:3 i" tDrJ "_-.. F- 35‘ 0.31 Era-\Aalole b. (4 points) A pseudo random number generator is given by Xi+1 = (6507 xi) mod 2‘“. (i) What is a maximum period? (ii) Can we achieve the maximum eriod with X = 2? p 0 5:2- 8 “AZ-2‘0 OJ‘0\. 67-0 / So moi‘iMuM .fo‘md (3:?— ..Q- m >10 as am 59 the ill penal it": :1 3x Kit '54-“. KO. A. . 7. A random variable has the following pdf: U24, 0 S x S 8 f(x)= 1/3, 8<x$10. 0, ofhenw'se Then the CDF is given as 3:! 24, 0 S x S 3 F(x) = { If3+(x—8)/3, 8 <x$10l Then its inverse CDF is given as 24R, 0 S R S a 3R + 7, b < R 51 ' What should be the values of a and b in the inverse CDF of X? Ciin 7' Vg ...
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This note was uploaded on 11/29/2009 for the course ISYE 3044 taught by Professor Alexopoulos during the Fall '08 term at Georgia Institute of Technology.

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PracticeTest1Soltn - Practice Test Solutions 1. (36 points)...

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