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Expressions - Probablity - Exam, Exams of Probability and Statistics

Aligarh Muslim UniversityProbability and Statistics

This is the Exam of Probablity which includes Joint Distribution, Continuous Random Variables, Compute, Test Engineer Discovered, Lifetime of an Equipment, Expected Lifetime, Variance, Parameter, Independent and Identically etc. Key important points are: Expressions, Simplifying, Probability, Failure Respectively, Redundant Array Of Independent, Inexpensive Disks, System, Helpful, Venn Diagram, Configuration

Typology: Exams

2012/2013

Uploaded on 02/20/2013

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2DI26 - Probability Theory
Final Exam
January 23rd, 2012
INSTRUCTIONS:
•This is a CLOSED NOTES exam. You are allowed only a CLEAN copy of the Statistical
Compendium and ONE SIDE OF ONE A4 SHEET with HANDWRITTEN notes.
•You can use a calculator (could be a graphical calculator). Cellphones, notebooks or
similar devices are not allowed. If you use any non-standard features of the calcula-
tor explain clearly how would you solve the question using only standard
features and/or the compendium, or you might not get full credit for your answer.
•There are 3 pages in the exam questionnaire (including this one) and you have 3 hours
(180 minutes) to complete the exam.
•The exam is graded on a scale from 0-100, and the final course grade takes also into
consideration the grade of the test of 15 of December. The allocation of points per
question is shown below
1 2 3 4 5
a b a b c d a b c d e f a b c d e a b
6666665655615565366
•The exam is to be done INDIVIDUALLY. Therefore discussion with your fellow col-
leagues is strictly forbidden.
•Please BE ORGANIZED IN YOUR WRITE-UP – we can’t grade what we can’t
decipher!
•A correct answer does not guarantee full credit, and a wrong answer does not guarantee
loss of credit. You should clearly and concisely indicate your reasoning and show all
relevant work. Your grade on each problem will be based on our best assessment of
your level of understanding as reflected by what you have written. JUSTIFY your
answers and be CRITICAL of your results.
•The problems are not necessarily in order of difficulty. I recommend that you quickly
read through all problems first, then do the problems in whatever order suits you best.
•Remember to IDENTIFY YOUR HANDOUT.
1
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2DI26 - Probability Theory

Final Exam

January 23rd, 2012

INSTRUCTIONS:

  • This is a CLOSED NOTES exam. You are allowed only a CLEAN copy of the Statistical Compendium and ONE SIDE OF ONE A4 SHEET with HANDWRITTEN notes.
  • You can use a calculator (could be a graphical calculator). Cellphones, notebooks or similar devices are not allowed. If you use any non-standard features of the calcula- tor explain clearly how would you solve the question using only standard features and/or the compendium, or you might not get full credit for your answer.
  • There are 3 pages in the exam questionnaire (including this one) and you have 3 hours (180 minutes) to complete the exam.
  • The exam is graded on a scale from 0-100, and the final course grade takes also into consideration the grade of the test of 15 of December. The allocation of points per question is shown below

1 2 3 4 5 a b a b c d a b c d e f a b c d e a b 6 6 6 6 6 6 5 6 5 5 6 1 5 5 6 5 3 6 6

  • The exam is to be done INDIVIDUALLY. Therefore discussion with your fellow col- leagues is strictly forbidden.
  • Please BE ORGANIZED IN YOUR WRITE-UP – we can’t grade what we can’t decipher!
  • A correct answer does not guarantee full credit, and a wrong answer does not guarantee loss of credit. You should clearly and concisely indicate your reasoning and show all relevant work. Your grade on each problem will be based on our best assessment of your level of understanding as reflected by what you have written. JUSTIFY your answers and be CRITICAL of your results.
  • The problems are not necessarily in order of difficulty. I recommend that you quickly read through all problems first, then do the problems in whatever order suits you best.
  • Remember to IDENTIFY YOUR HANDOUT.

P.I: (12 points) Evaluate the following expressions, simplifying your result as much as pos- sible. (a)

∫ (^) e 2

e

x ln x

dx (b)

t cos(t)dt

P.II: (24 points) A RAID system (Redundant Array of Independent/Inexpensive Disks) was built using 3 disks, identified with the names disk1, disk2, and disk3. These disks are known to have probability of failure respectively 0.01, 0.03, and 0.05. The disks are also known to fail independently. Let A denote the event {disk1 failure}, and similarly B and C denote the events disk and disk3 failure, respectively.

(a) The RAID system is such that there is loss of data only if two or more disks fail. Let E denote this event. Write E as a function of events A, B and C, using set notation. (Hint: you might find helpful to draw a Venn diagram). (b) Compute the probability there is loss of data, that is, compute P (E). (Hint: write the event E as the union of four mutually exclusive events). (c) Due to a mistake in the configuration of the system there will instead be loss of data if at least one of the following happens: (i) disk1 fails; (ii) disk2 and disk both fail. What is now the probability that there is a loss of data? (d) Consider the setting of question (c). Given that disk 3 has failed, what is now the probability there will be loss of data?

TIP: For the entire problem it might be helpful to recall the following important facts: Let A and B be two independent events. Then event A is also independent of B′, and event A′^ is also independent of B and of B′.

P.III: (28 points) The number of failures that occur in a certain computer network is well described by a Poisson process. On average, the time between two consecutive failures is 16 hours.

(a) Let Y be the number of failures that occur in one day (24 hours). What is the distribution of Y? Compute P (Y ≥ 2). (b) What is the probability that, over the period of one week (7 days), there is exactly one day without any network failures? (c) Let M be the total number of network failures that occur in a month (30 days). Evaluate P (M > 40) using the normal approximation.