1. Introduction Random Process

2. Where do we start from? Undergraduate Graduate Probability course Our main course Review and Additional course If we have enough time

3. PART I INTRODUCTION SYSTEM MODEL THEORY OF PROBABILITY EXAMPLE: COMMUNICATION OVER UNRELIABLE CHANNELS

4. Why do we have to study random process?  Wireless communication networks provide voice and data transfer in severe interference environments  The vast majority of media signal, voice, audio, images, and video are processes digitally  Huge Web server farms deliver vast amounts of highly specific information to users

5. Mathematical Model  Model: an approximate representation of a physical situation  Mathematical model: used when the observational phenomenon has measureable properties  Consists of a set of assumptions about how a system or physical process works  Stated in the form of mathematical relations involving the important parameters and variables of the system  Computer simulation model: consists of a computer program that simulates or mimics the dynamics of a system

6. Modeling Process

7. Types of Model  Deterministic model  The conditions under which an experiment is carried out determine the exact outcome of the experiment  Example: circuit theory models that consist of Kirchhoff’s voltage and current law and also Ohm’s law  Probability models  Involve phenomena that exhibit unpredictable variation and randomness  Random experiment: experiment in which the outcome varies in an unpredictable fashion when the experiment is repeated under the same conditions  Example: outcome of tossing a coin

8. Probability model  Example: taking an identical ball labeled 0, 1, and 2

9. Probability model  Statistical regularity: in order to make prediction, the system must exhibit regularity, i.e. It must have averages obtained in long sequence of repetitions  Relative frequency:  When, the parameters is called probability Number of repetition Number of outcome k

10. Probability model

11. The conditions under which a random experiment is performed determine the probabilities of the outcomes of an experiment

12. Probability model  Propertive of relative frequency: We definitely know that By dividing the equation by n, we get The sum of the number of occurence must be Therefore,

13. Theory of Probability

14. Example  We take an example: communication over unreliable channels

15. PART II BASIC CONCEPTS OF PROBABILITY THEORY

16. What should you do yourself?  Please review:  Random experiments – sample space – events – set theory  The axioms of probability  Conditional probability  Bayes’ rule  Independence of events However, let me remind of you the formulas and some examples

17. Review of set theory

18. Review of axioms of probability

19. Conditional probability

20. Homeworks All problems are from Garcia’s book, due date: next week For International Class:  Please do problems 2.2, 2.21, 2.62 For Regular Class:  Please do three problems. The problems depend on your absent number. If your absent number is n, please do problems 2.n, 2.(22 + n), and 2.(44 + n)  Example: if your absent number is 1, do problems 2.1, 2.23, and 2.45  If your absent number is 22, do problems 2.22, 2.44, and 2.66