Lecture 1 introduction

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Lecture 1 introduction

  1. 1. MTH-263 UndergraduateProbability Theory and Random Variables Lecture 1 Introduction and Basic Ideas Muhammad Adnan Siddique masiddique@ciitlahore.edu.pk Computer Vision Research Group (COMVis) Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore Department of Electrical Engineering, CIIT Lahore
  2. 2. Course Organization• Course Code: MTH-263• Credits: 3 (2 lectures per week)• Student Batch: FA10-BTE-A,B, FA10-BCE• Lecture Schedule: to be decided yet• Evaluation: Assignments/Quizzes/Evaluations/Sessional & Terminal Exam• Instructor Email: masiddique@ciitlahore.edu.pk• Visiting Hours: please send an email first 2 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  3. 3. Resource Materials/Books• [ALG] Alberto Leon-Garcia. Probability and Random Processes for Electrical Engineers. 2nd ed. Pearson Education, Low price edition. ISBN: 8129703173.• [Walp] Ronald E. Walpole. Probability and Statistics for Engineering and Scientists. Pearson Education, Low price edition.• [OWB] Athanasios Papulis, S. U. Pillai. Probability, Random variables and Stochastic Processes. Tata McGraw-Hill• Lecture slides (for class-room learning) 3 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  4. 4. Moodle Settings …• Visit the link: http://mycourseportal.info/ 4 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  5. 5. Moodle Settings …• Visit the link: http://mycourseportal.info/ 5 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  6. 6. Moodle Settings …• Visit the link: http://mycourseportal.info/ 6 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  7. 7. Moodle Settings …• Visit the link: http://mycourseportal.info/ 7 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  8. 8. Software• For self-learning, implementation of the ideas learnt in the course, project-work etc.• MATLAB or IDL (Proprietary software)• Many freeware, open-source software available, e.g. • FreeMAT (a kind of mini-MATLAB) • Octave 8 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  9. 9. Introduction Probability Theory Stochastic Signal Processing Random Variables and Distributions Random/Stochastic Signal Processing ProcessesRequisite of numerous other courses:• Wave Propagation/Wireless Communication• Communication Theory• Information Theory• Pattern Recognition• Radar and Sonar Signal Processing, etc. 9 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  10. 10. Randomness• Randomness of phenomena • Chaos • Uncertainty • Doubt• We ‘certainly’ desire some level of ‘certainty’ in our lives!!!• And why so? • Traffic is chaotic, but nobody wants to get stuck! • Weather is uncertain, nobody wants the flights getting cancelled!• And we are engineers! , which makes us special ! • We understand numbers!• We need to quantify ‘certainty’! 10 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  11. 11. Models of a Physical Situation • A model is an approximate representation. • Mathematical model • Simulation Models • Deterministic versus Stochastic/Random  Deterministic models offer repeatability of measurements • Ohm’s Laws, model of a capacitor/inductor/resistor  Stochastic models don’t: • Processor’s caching, queuing, and estimation of task execution time • The emphasis of this course would be on Stochastic Modeling. 11 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  12. 12. Random Experiment• Random Experiment: the outcome of the experiment varies in an random manner• Sample Space: the set of possible results of the experiment, without repetition• Outcome: an entity of the sample space, a singular result of the experiment• Consider an urn with three balls in it, labeled 0,1,2 • What are the chances that a ball withdrawn at random from the urn is labeled ‘1’? • How to quantify this ‘chance’? 1 2 3 12 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  13. 13. The urn experiment … Questions?• How to quantify this ‘chance’?• Is withdrawing any of the three balls equally likely (equi-probable); or if any ball is more likely to be drawn compared to the others?• If someone assigns that ‘1’ means ‘sure occurrence’ and ‘0’ means ‘no chance of occurrence’, then what number would you give to the chance of getting ‘ball 1’?• And how do you compare the chance of withdrawing an odd-numbered ball to that of withdrawing an even-numbered ball? Lets see how Nature answers? 1 2 3 13 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  14. 14. The urn experiment … 14 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  15. 15. The urn experiment …• Counts of the selections of ‘kth’ outcome in ‘n’ iterations (trails) of the random experiment is given by• The relative frequency of ‘kth’ outcome is then defined as: 15 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  16. 16. The urn experiment … 100 iterations 16 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  17. 17. The urn experiment … 1000 iterations Statistical regularity Probability defined as the limiting case of relative frequency 17 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  18. 18. Inferences• Statistical regularity: long-term averages of repeated iterations of a random experiment tend to yield the same value• A few ideas of note: • And regarding the chances of withdrawing an odd-numbered ball, • Take note: these ideas are only as inculcated by nature; therefore, our modeling of problems should never contradict them 18 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore
  19. 19. Axiomatic Approach and Probability Assignment• The random experiment is defined and a set of all possible outcomes, S is obtained• Events are defined. An event may comprise of one or more outcomes.• Probabilities (which are numbers) are assigned to each outcome/event, on the basis of the following axioms based on the previous inferences, for any event A,  P[A] >= 0 (which means probability is never negative)  P[S] = 1 (which means that the total probability is always 1, and any event or outcomes probability is never more than 1)  For two events A and B which do not occur simultaneously, P[A or B] = P[A] + P[B] 19 Computer Vision Research Group (COMVis) Department of Electrical Engineering, CIIT Lahore

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