ESE503 - Simulation Modeling & Analysis (Homework #10) University of Pennsylvania ESE 503

ESE503 - Simulation Modeling & Analysis (Homework #10)
Spring Semester, 2022 M. Carchidi
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Problem #1 (20 points)
The lifetimes (in hours) of 20 devices have been measured and the results are shown in the
table below.
# Lifetime # Lifetime # Lifetime # Lifetime
1 2.494 6 2.715 11 6.434 16 0.713
2 1.296 7 5.922 12 0.718 17 0.199
3 4.350 8 1.190 13 2.259 18 3.929
4 0.271 9 0.391 14 0.197 19 2.934
5 0.515 10 0.520 15 0.242 20 2.721
Construct a quantile-quantile plot of the data assuming an exponential distribution with
mean equal to the sample mean. Compute the best-fit linear regression line and plot it on
the quantile-quantile graph.
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Problem #2 (15 points)
The pdf of the Rayleigh distribution with parameter   0 is given by
1 2
fx;   xe − 2 x for 0  x.
If a random sample X 1 , X 2 , X 3 , … , X n  is chosen from this distribution,
a.) (8 points) determine an expression (in terms of X 1 , X 2 , X 3 , … , X n and n), for the maximum
likelihood estimate for the parameter  (call it  ML ).
b.) (7 points) Determine an expression (in terms of X 1 , X 2 , X 3 , … , X n and n), for the estimate
you would get by just fitting the expected value of the above distribution to the sample
mean (call it  SM ). You may use the fact that

0 u2e− 1
2
u2
du   .
2
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Problem #3 (20 points)
Suppose that X is a geometric random variable with parameter   0, so that the pmf and
cdf of X are
px  1 −  x−1  and Fx  1 − 1 −  x ,
respectively, for x  1, 2, 3, … . A sample X 1 , X 2 , X 3 , … , X n  from X is collected and a
statistic S, defined by
S  minX 1 , X 2 , X 3 , … , X n 
is computed. Determine the p-value of S if the following sample
X 1 , X 2 , X 3 , X 4 , X 5   4, 2, 3, 5, 2
of X is obtained, given also that   0. 153.
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Problem #4 (15 points)
Records pertaining to the monthly numbers of job-related injuries at an underground coal
mine were being studied by a federal agency. The values for the past 100 months were as
follows.
Injuries Per Month Frequency of Occurrence
0 35
1 40
2 13
3 6
4 4
5 1
≥6 1
a.) (5 points) Apply a chi-square test to these data to test the hypothesis that the underlying
distribution is Poisson assuming a level of significance of   0. 05.
b.) (5 points) Apply a chi-square test to these data to test the hypothesis that the underlying
distribution is Poisson with a mean of 1.0, assuming a level of significance of   0. 05.
c.) (5 points) What are the differences (if any) in parts (a) and (b), and when might each case
arise?
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Problem #5 (15 points)
A random sample 0. 16, 0. 77, 0. 48, 0. 69, 0. 21, of size n  5, is taken from a continuous
random variable X having pdf
  2 −1
fx,   x 1 − x 2 
2
for 1   and 0 ≤ x ≤ 1.
a.) (8 points) Determine the resulting maximum likelihood estimate for . Hint: You may use
the fact that
dx −1 
 x −1 lnx for 1  .
d
b.) (7 points) Determine the estimate for  if the mean of the distribution is to equal the
sample mean of the data. Hint: You may use the fact that
1
 0 x n dx  n 1 1
for n ≠ −1.
Of course, recall the fact that the solution to the equation
−B  B 2 − 4AC
Az 2  Bz  C  0 is z
2A
for A ≠ 0.
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Problem #6 (15 points)
The time (in minutes) between requests for the hookup of electric service was accurately
maintained at the Gotwatts Flash and Flicker Company with the following results for the
last 50 requests.
0.661 4.910 8.989 12.801 20.249
5.124 15.033 58.091 1.543 3.624
13.509 5.745 0.651 0.965 62.146
15.512 2.758 17.602 6.675 11.209
2.731 6.892 16.713 5.692 6.636
2.420 2.984 10.613 3.827 10.244
6.255 27.969 12.107 4.636 7.093
6.892 13.243 12.711 3.411 7.897
12.413 2.169 0.921 1.900 0.315
4.370 0.377 9.063 1.875 0.790
How are the times between requests for service distributed? Develop and test a suitable
model.
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Simulation Problem #8 - One More Monte-Carlo Problem
a.) (10 points) Suppose you have two coins of radii R 1 and R 2 , respectively. The coins are to
be tossed onto the xy plane so that the centers of each coin are given by (X 1 , Y 1 ) and
(X 2 , Y 2 ), respectively. If X 1 , Y 1 , X 2 and Y 2 are all independent and identical such that
X 1 , Y 1 , X 2 , Y 2  N,  2 . Compute in terms of R 1 , R 2 and , the probability that the two
coins are not touching in any way and then run a Monte-Carlo simulation (1000 trials)
using different values of R 1 , R 2 and  to test your analytic result. Note that your final
answer should just involve the ratio
  R1  R2 .
 2
b.) (10 points) Suppose you have two spherical bubbles of radii R 1 and R 2 , respectively. The
bubbles are to be tossed into xyz space so that the centers of each bubble is given by
(X 1 , Y 1 , Z 1 ) and (X 2 , Y 2 , Z 2 ), respectively. If X 1 , Y 1 , Z 1 , X 2 , Y 2 and Z 2 are all independent
and identical such that X 1 , Y 1 , Z 1 , X 2 , Y 2 , Z 2  N,  2 . Compute in terms of R 1 , R 2 and ,
the probability that the two bubbles are not touching in any way and then run a
Monte-Carlo simulation (1000 trials) using different values of R 1 , R 2 and  to test your
analytic result. Note that your final answer should just involve the ratio
  R1  R2 .
 2
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