# ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 1 - General info, Sets and Probabilistic Models Farinaz Koushanfar ECE Dept., Rice University

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• Slide 1
• ELEC 303, Koushanfar, Fall09 ELEC 303 Random Signals Lecture 1 - General info, Sets and Probabilistic Models Farinaz Koushanfar ECE Dept., Rice University Aug 25, 2009
• Slide 2
• ELEC 303, Koushanfar, Fall09 General information Syllabus/policy handout Course webpage: http://www.ece.rice.edu/~fk1/classes/ELEC303.htm Required textbook: Dimitri P. Bertsekas and John N. Tsitsiklis. Introduction to Probability. 2 nd Edition, Athena Scientific Press, 2008Introduction to ProbabilityAthena Scientific Press Recommended books on webpage Instructor and TA office hours on the webpage
• Slide 3
• ELEC 303, Koushanfar, Fall09 Grading / policy Grading Quiz 1: 10% Midterm: 20% Quiz 2: 10% Final: 30% Homework: 15% Matlab assignments: 10% Participation: 5% Read the policy handout for the homework policy, cheating policy, and other information of interest
• Slide 4
• ELEC 303, Koushanfar, Fall09 Lecture outline Reading: Sections 1.1, 1.2 Motivation Sets Probability models Sample space Probability laws axioms Discrete and continuous models
• Slide 5
• ELEC 303, Koushanfar, Fall09 Motivation What are random signals and probability? Can we avoid them? Why are they useful? What are going to learn?
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• ELEC 303, Koushanfar, Fall09 Sets quick review A set (S) is a collection of objects (x i ) which are the elements of S, shown by x i S, i=1,,n S may be finite or countably infinite
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• ELEC 303, Koushanfar, Fall09 Sets quick review Set operations and notations: Universal set: , empty set: , complement: S c Union: Intersection: De Morgans laws: 1 2
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• ELEC 303, Koushanfar, Fall09 Venn diagram A representation of sets A AcAc A B A B A B C A B C S T U
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• ELEC 303, Koushanfar, Fall09 Probabilistic models Sample space: set of all possible outcomes of an experiment (mutually exclusive, collectively exhaustive) Probability law: assigns to a set A of possible outcomes (events) a nonnegative number P(A) P(A) is called the probability of A Figure courtesy of Bertsekas&Tsitsiklis, Introduction to Probability, 2008
• Slide 10
• ELEC 303, Koushanfar, Fall09 From frequency to probability (1) Recovery time Age Slide courtesy of Prof. Dahleh, MIT The time of recovery (Fast, Slow, Unsuccessful) from an ACL knee surgery was seen to be a function of the patients age (Young, Old) and weight (Heavy, Light). The medical department at MIT collected the following data:
• Slide 11
• ELEC 303, Koushanfar, Fall09 From frequency to probability (2) What is the likelihood that a 40 years old man (old!) will have a successful surgery with a speedy recovery? If a patient undergoes operation, what is the likelihood that the result is unsuccessfull? Need a measure of likelihood Ingredients: sample space, events, probability Slide courtesy of Prof. Dahleh, MIT
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• ELEC 303, Koushanfar, Fall09 Sequential models sample space Slide courtesy of Prof. Dahleh, MIT 2 1 3
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• ELEC 303, Koushanfar, Fall09 Axioms of probability 1.(Nonnegativity) 0P(A)1 for every event A 2.(Additivity) If A and B are two disjoint events, then the probability P(A B)=P(A)+P(B) 3.(Normalization) The probability of the entire sample space is equal to 1, i.e., P( )=1
• Slide 14
• ELEC 303, Koushanfar, Fall09 Discrete models Example: coin flip head (H), tail (T) Assume that it is a fair coin What is the probability of getting a T? What is the probability of getting 2 Hs in three coin flips? Discrete probability law for a finite number of possible outcomes: the probability of an event is the sum of its disjoint elements probabilities
• Slide 15
• ELEC 303, Koushanfar, Fall09 Example: tetrahedral dice Let every possible outcome have probability 1/16 P(X=1)=P(X=2)=P(X=3)=P(X=4)=0.25 Define Z=min(X,Y) P(Z=1)=? P(Z=2)=? P(Z=3)=? P(Z=4)=? 2 1 3 Discrete uniform law: Let all sample points be equally likely, then Example courtesy of Prof. Dahleh, MIT
• Slide 16
• ELEC 303, Koushanfar, Fall09 Continuous probability Each of the two players randomly chose a number in [0,1]. What is the probability that the two numbers are at most apart? Draw the sample space and event area Choose a probability law Uniform law: probability = area x y 1 1 Example courtesy of Prof. Dahleh, MIT
• Slide 17
• ELEC 303, Koushanfar, Fall09 Some properties of probability laws If A B, then P(A) P(B) P(A B) = P(A) + P(B) - P(A B) P(A B) P(A) + P(B) P(A B C) = P(A) + P(A c B) + P(A c B c C)
• Slide 18
• ELEC 303, Koushanfar, Fall09 Models and reality Probability is used to model uncertainty in real world There are two distinct stages: Specify a probability law suitably defining the sample space. No hard rules other than the axioms Work within a fully specified probabilistic model and derive the probabilities of certain events