MATH 401 Probability and Statistics for IET and MET 4th Semester

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MATH 401Probability and Statistics for IET and
MET 4th Semester

• Spring 2009

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Class Meetings

• MET Group
• When Monday _at_ 1400
• Where H13
• IET Group
• When Monday _at_ 830
• Where H13

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Teaching Staff

• Dr. Oleksiy Us - Lecturer
• Tutors
• Dr. Ahmed Mostafa
• Mr. Mostafa Ogeil
• Mr. Tarek Mounir

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Lecturer

• Dr. Oleksiy was born in Ukraine.
• Ukraine
• was a part of Soviet Union
• is independent since 17 years
• qualified for the Football World Cup 2006, and
  reached the quarterfinals.

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Office Hours

• Try my office (currently C5.303)
• OR
• Make an appointment via
• E-mail oleksiy.us_at_guc.edu.eg

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Course on the Intranet

• All the course materials are collected in a
  special folder on the Intranet.
• The folder contains subfolders with lectures,
  solutions to home work problems, course schedule,
  announcements, etc.
• Check it regularly you might find useful
  information there!

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Assessment System

• 3 Quizzes 15
• Assignments 15
• Midterm 30
• Final Exam (comprehensive, with higher emphasis
  on the second part
• of the course) 40

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Assessment System. Quizzes

• The details will be given to you by your
  instructors.
• The GUC rule applies best 2 out of 3 quiz
  results will be taken into consideration.

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Home Work

• A number of exercises will be given as homework
  every week (see Course Plan).
• Homework is set to
• give you practice in applying the concepts
  covered in the course
• give you a chance to assess the level of your
  understanding.

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Class Work Assignments

• In every class, you will receive a work sheet
  with most important exercises on the weeks
  topic. These examples will be discussed in class.
• Some problems on the sheet will be left as an
  exercise (in class or at home).
• An assignment sheet with a couple of questions
  will be attached to each work sheet. The
  questions are to be done at home and submitted a
  week later. This work will be graded.

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Attendance of Lectures

• is not compulsory, but
• Tutorial work sheets do not repeat or replace
  lectures. They only complement the lectures by
  giving the students a chance to communicate with
  their tutors and to practice under supervision.
• Tutorials do not cover all the material that may
  appear in the midterm and the final exams.

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Students Comments

• MCQs in the final exam were rather tricky
  but it is because we depend on the tutorials all
  the time
• The final was very difficult
• Not fair exams

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Students Comments

• He gave us the wrong sense at the beginning of
  the semester that the course was going to be
  relatively easy, which it is not all the way.

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Conclusion

• Lecture materials are a must if you aim for a
  high grade.

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Lecture Notes

• Slides to all lectures will be available on the
  intranet.
• In addition, notes to Lectures 1-8 will be
  available as PDF files.

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Lecture Notes

• Lecture Notes summarize all important issues that
  were discussed in lectures, and contain important
  examples.
• However, the only complete source for this course
  is a text-book.

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Textbooks

• Douglas C. Montgomery, George C. Runger. Applied
  Statistics and Probability for Engineers (3rd
  ed.). John WileySons
• S.Ross. Probability and Statistics for Engineers
  and Scientists (3rd ed.). Elsevier AP. (More
  challenging!)

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Course Description

• This course introduces students to the
  fundamentals of the probability theory and the
  basics of statistical analysis.
• For a detailed course description and the list of
  learning objectives see the Course Outline on the
  intranet.

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Question 1

• On average, one request in 1000 causes a server
  to crash.
• What are the chances that one server can process
  2000 requests or more?
• If you want the system to process 10000 requests
  without crashing, 99.99 of the time, how many
  servers should be connected, so that the second
  server supports the system when the first one
  breaks down, etc.?

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Question 2

• Suppose that 3600 messages are received within
  one hour.
• What are the chances that one message per second
  is received?
• What are the chances that the time between two
  consecutive messages is less than one second?

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Question 3

• Ten measurements of the current at a certain
  resulted in an average value of 123 mA. You
  expected, however, a value not greater than
  120mA.
• So was your assumption wrong? Could it be
  correct?
• How would the answer be affected, if twenty
  measurements were carried out?
• What other factors are to consider?

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Course Structure

• Fundamentals of Probability Theory Conditional
  Probability and Independent Events. Random
  Variables. Central Limit Theorem. (8 Lectures)
• Basic Notions of Statistics. Descriptive
  Statistics Organizing, Presenting and
  Summarizing Data. (1 Lecture)
• Inferential Statistics Parameter Estimation,
  Hypothesis Testing. (3 Lectures)

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Introduction to Probability Theory

• Lecture 1

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Probability Experiments

• A random experiment is an experiment that can
  result in different outcomes, even though it is
  repeated in the same manner every time.

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Probability Experiments

• An experiment
• Flipping a coin once.
• Rolling a die once.
• Pulling a card from a deck.

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Further Examples

• More relevant examples are
• An experiment
• Checking a transmitted bit on error.
• Transmitting bits until the first error.
• Measuring the time between two consecutive
  requests to a server.
• Measuring the thickness of a wafer used in
  semiconductor manufacturing.

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Sample Spaces

• The set of all possible outcomes of a random
  experiment is called a sample space.
• You may think of a sample space as the set of all
  values that a variable may assume.
• We are going to denote the sample space by S.

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Examples

• Experiment Tossing a coin once.
• S H, T
• Experiment Rolling a die once.
• S 1, 2, 3, 4, 5, 6
• Experiment Drawing a card.
• S classic-playing-cards.png

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Examples

• Experiment Checking a bit.
• S o, e
• Experiment Transmitting until the first error.
• S e,oe,ooe,oooe,ooooe,
• Experiment Measuring the time.
• S 0,T, or R.

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Classification of Sample Spaces

• We distinguish between discrete and continuous
  sample spaces.
• The outcomes in discrete sample spaces can be
  counted. The number of outcomes can be finite or
  infinite.
• In the case of continuous sample spaces, the
  outcomes fill an entire region in the space
  (interval on the line).

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Events

• An event, E, is a set of outcomes of a
  probability experiment, i.e. a subset in the
  sample space.
• E ? S.
• If an event E contains no outcomes, then E is an
  impossible event.
• The union and the intersection of events are
  defined in a natural way.

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Mutually Exclusive Events

• Two events, E and F, are called mutually
  exclusive, if
• That is, it is impossible for an outcome to be an
  occurrence of both events.

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Mutually Exclusive Events

• When drawing a card
• Drawing a club and drawing a heart are mutually
  exclusive events.
• Drawing a club and drawing a king are not
  mutually exclusive
• Drawing the king of clubs is an occurrence of
  both events.

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Examples

• Experiment Rolling two dice.
• S (1,1), (1,2), . . . . . , (1,6),
• (2,1), (2,2), . . . . , (2,6),
• ...
• (6,1), (6,2), , (6,6)

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Example

• E is the event of getting two numbers with
  different parities.
• F is the event of getting two numbers whose sum
  is 6.
• E (1,2), (1,4), (1,6),
• (2,1), (2,3), (2,5),
• ..,
• (6,1), (6,3), (6,5)
• F (1,5), (2,4), (3,3), (4,2), (5,1)
• Thus, E and F are mutually exclusive.
• For the sum of two number to be even, they have
  to have the same parity.

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Complementary Events

• The complement of an event E is defined by
• where \ denotes set difference. That is the
  complement contains all outcomes that are not in
  E.
• It is clear that E and E are mutually exclusive.

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Venn Diagrams

• Sometimes it is convenient to represent the
  sample space by a rectangular region, with events
  being circles within the region. Such a
  representation is called Venn diagrams.

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De Morgan Laws

• Using the Venn diagrams one can easily show that
• (1st Law)
• The second rule can be obtained from the first
  one by replacing E and F by E and F,
  respectively, in the left-hand side of the
  equation.

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Definition of Probability

• The probability of an event E is a number, P(E),
  such that
• if the events E and F are mutually exclusive.
• The latter implies that, in general,

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Further Properties of Probability

• Probability of sub-event
• Probability of the union of three events

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Classical Probability

• The (classical) probability, P(E), of an event E
  is given by
• where N() stands for the number of elements in a
  set.
• It is assumed here that the sample space is
  finite and all outcomes are equally likely to
  occur.

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Computing Probabilities

• Find the probability that a 1-byte long message
  contains exactly 4 zeros.
• If a coin is flipped 8 times, find the
  probability that heads appear exactly four times.

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Computing Probabilities

• It is clear that an efficient way to count
  outcomes in large sample spaces is required.
• So next weeks topic is
• COUNTING RULES

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Class Meeting Next Week

• IET Group
• When Wednesday _at_ 830
• Where H14
• MET Group
• When Thursday _at_ 830
• Where H11

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Thank you