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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(2X3Y)=? 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 30min ute intervals, and jobs will be
processed ﬁrstoome—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) OéQQUS 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 30minute intervals, and jobs
will be processed ﬁrstcomeﬁrstserved. 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 jobtype inputs to a simulation of this system can be
generated by rolling a fair and sixsided die. That is. Pr{JobType = j} =1 j=1,2,...,6. Derive the inverse—CDF that 6 3
generates the jobtype inputs. b. (4 points) Generate 4 jobtype inputs when R = 0.1. 0.5, 0.6. 0.9
using the inverseCDF 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: {VIIt’ll“) Mad3 = 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 chisquared 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  Agod A \‘l
(d) True 0 The covariance is a unitfree measure of dependency .(Tar tr “I‘dUH?)
between two random variables. Corf'e\o\—\or\ ts amt “Free (e) True oTwo random variables X and Y are independent if the
productmoment 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 “AZ2‘0 OJ‘0\. 670 / 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.
 Fall '08
 ALEXOPOULOS

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