notes02 - TELENET 2120: NETWORK PERFORMANCE Basic...

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1 TELENET 2120: NETWORK PERFORMANCE Basic Probability Asst. Prof. Joseph Kabara, Ph.D Graduate Program in Telecommunications and Networking University of Pittsburgh 2 Spring 2008, Class #2 TELENET 2120: Network Performance Terminology Probability Theory - based on the concept of a random experiment Random – phenomenon/experiment where an individual outcome is uncertain but there is a regular distribution of outcomes in a large number of repetitions. Probability - proportion of times a specific outcome would occur in a long series of repetitions of the experiment. 3 Spring 2008, Class #2 TELENET 2120: Network Performance Long-Term Relative Frequency – If toss a single coin, the relative frequency of heads is erratic for 2, or 5, or 10 tosses. – If you toss the coin several thousand times, the relative frequency remains stable. Mathematical probability is an idealization of what would happen to the relative frequency after infinite number of repetitions of random experiment. n repetition n in occurs A event times of number frequency relative = What is Probability?
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2 4 Spring 2008, Class #2 TELENET 2120: Network Performance Probability of Heads Probability based on long term term relative frequency! 0.501 501 1000 0.55 11 20 0.60 3 5 0.33 1 3 1.00 1 1 Relative Frequency # Heads # Tosses 5 Spring 2008, Class #2 TELENET 2120: Network Performance Probability Model Sample Space - set of all possible outcomes of a random experiment ( S ). Event - an outcome or a set of outcomes of the experiment Probability measure is a number or function that maps from the events in the sample space to a real number between 0 and 1 The probability of all possible outcomes (that is the sample space) must equal 1 6 Spring 2008, Class #2 TELENET 2120: Network Performance Probability Model Example: Toss of a single die Sample Space : S = {1,2,3,4,5,6} Event : A = {rolled even number}, B = {rolled odd number}, D= {rolled a 2} Probability measure: P(A) = .5, P(B) = .5, P(D) = 1/6
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3 7 Spring 2008, Class #2 TELENET 2120: Network Performance Probability Rules Remember the probability of any event P(A) must satisfy 0 < P(A) < 1 Complement Rule The complement of any event A is the chance that A does not occur P(A c ) = 1 - P(A) Example: Toss a single die: S = {1,2,3,4,5,6}; let A = {2,4}, A c = {1,3,5,6}; P(A) = 1/3; P(A c ) = 1-1/3 = 2/3 Addition Rule For two events A and B that are disjoint (no common outcomes) P (A or B) = P(A) + P (B) Example: Toss a single die: S = {1,2,3,4,5,6}; let A = {2}, B = {1,3,5}; P(A or B) = P(A) + P(B) = 1/6 + 1/2 = 2/3 8 Spring 2008, Class #2 TELENET 2120: Network Performance Probability Rules Multiplication Rule = two events A and B are independent, if knowing that one occurs does not change the probability that other occurs P (A and B) = P(A)*P(B) Example: Toss a pair of die . S = {(1,1),(1,2),….(6,6)} 36 possible outcomes. Let A ={first die shows 6} = {(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)} Let B = {second die shows 1} = {(1,1),(2,1),(3,1),(4,1),(5,1),(6,1)} Then P(A) = 6/36 = 1/6; P(B) = 6/36 = 1/6 and P(first die 6, second die 1) = P(A and B) = 1/36 = P(A) P(B) implies independence 9 Spring 2008, Class #2 TELENET 2120: Network Performance Probability Rules Multiplication Rule Example of Dependent Case: Toss a pair of die S = {(1,1),(1,2),….(6,6)} ; 36 possible outcomes
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notes02 - TELENET 2120: NETWORK PERFORMANCE Basic...

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