Introduction Random Process
Where do we start from? Undergraduate Graduate Probability course Our main course Review and Additional course If we have enough time
PART I INTRODUCTION SYSTEM MODEL THEORY OF PROBABILITY EXAMPLE: COMMUNICATION OVER UNRELIABLE CHANNELS
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
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
Modeling Process
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
Probability model Example: taking an identical ball labeled 0, 1, and 2
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
Probability model
The conditions under which a random experiment is performed determine the probabilities of the outcomes of an experiment
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,
Theory of Probability
Example We take an example: communication over unreliable channels
PART II BASIC CONCEPTS OF PROBABILITY THEORY
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
Review of set theory
Review of axioms of probability
Conditional probability
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