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CEE11 Midterm Exam: Analysis of Uncertainty
Spring 2001 Thursday May 10
Name:__________________________ Student #:________________________________
7 questions – 75 minutes – please read carefully – turn your paper over when you have finished
Instructions:
1.
This is a closed book exam
2.
There are seven problems and seven pages for this exam (including cover sheet and extra sheet for
calculations). Check your copy and alert instructor if any pages are missing.
3.
Please read all questions carefully!
4.
SHOW ALL WORK! Partial credit will be given if all work, assumptions and reasoning are clearly
explained.
Instructor’s use
1.______________(10%)
2.______________(10%)
3.______________ (5%)
4.______________ (5%)
5.______________ (15%)
6.______________ (25%)
7.______________ (30%)
Total
________________
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Question 1) (10%; estimated time - 5 minutes)
Average GRE scores for incoming graduate students in Engineering at UCI are examined by country of
origin to determine whether trends in performance differ.
The long run average for Portuguese students is 750 on the mathematics portion, based on a rather small
sample of 17. If this year’s incoming Portuguese students scored, 710, 730 and 770, recalculate the long
run average for Portuguese students.
Question 2) (10%; estimated time - 5 minutes)
Consider a sample x1, x2, ...xn , and suppose that the values of
a) Let
x , S x2 and sx have been calculated.
yi  xi  x . Calculate y , S y2 and s y in terms of x , S x2 and sx
y  ________________
S y2 = _________________ s y  __________________________
Question 3) (5% estimated time – 5 minutes)
Events A, B and C have associated probabilities P( A)  0.5 , P( B)  0.6 . Further P( A  B)  0.2
What is P( B  A) ? ______________________________
What is P( B | A) ? ______________________________
What is P( A | B) ? ______________________________
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Question 4) (5%; estimated time – 5 minutes)
a) How many different 6 digit license plates can be constructed (only English letters and numbers, no
symbols) if each license plate must have exactly 6 characters?
Question 5) (15%; estimated time - 10 minutes)
An engineering professor wishes to schedule appointments with each of her 10 research assistants, 3 PhD
students, 3 MS students and 4 undergraduates.
a)
In how many ways could she select 4 of these students to meet with on Wednesday?
b)
What is the probability that at least one undergraduate will be assigned a Wednesday meeting?
c)
What is the probability that all three PhD students will be selected?
3
Question 6) (20%; estimated time - 20 minutes)
Consider a system of components configured in the following way.
A=
0.90
B =0.90
C=0.50
D=0.80
E=0.50
The lower component, made up of parts C, D and E functions only if all three of these function while the
whole system functions if either A, B or CDE function. Parts A, B and CDE are mutually independent –
failure of one part does not affect the others. Parts C, D and E are also mutually independent of each other.
a)
If the numbers listed on the components represent reliability on a given day, what is the
probability that part CDE will function on that day?
b)
If the numbers listed on the components represent reliability on a given day, what is the
probability that part CDE will not function on that day?
c)
What is the probability that both parts A and B will function?
d)
What is the probability that the whole system will function? Hint: a three circle Venn diagram
should help if you get stuck.
4
Question 7) (30%; estimated time - 15 minutes)
A simple random walk is the following:
A
S
B
D
C
a)
A person begins at S (the starting point) and flips a coin and moves either right or left. Then he
flips another coin and moves right or left again. If he finds himself at A or D, he returns immediately back
his starting point.
Letting X represent the number of moves made (links traversed) in trips beginning and ending with S.
Please note that the links from D or A to S are to be counted.
Derive the probability mass function of X.
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b)
Now assume we have the following (slightly more complicated) network. The person begins at
H and then selects one of the three non-home destinations at random. Then, he selects from the two
adjacent non-home destination or his home, again, with all three destinations having equal probability.
Letting Y be the number of visits to any of the non-home destinations in trips beginning and ending at
H, derive the probability mass function of Y.
A
B
H
C
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Extra Sheet of Paper
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