Lecture2_Spring11

Lecture2_Spring11 - Probability Model Axiomatic approach...

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Probability Model Elec210 Lecture 2 1 Practical Problem Probability Model Random Experiments Axiomatic approach
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Elec210 Lecture 2 2 Elec 210: Lecture 2 Specifying Random Experiments Sample spaces and events Set Operations The Three Axioms of Probability Corollaries Probability Laws for Picking Events at Random Birthday Problem
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Elec210 Lecture 2 3 Specifying Random Experiments A random experiment is specified by stating an experimental procedure and one or more measurements or observations. The sample space S is the set of all possible outcomes. An outcome is a result that cannot be decomposed into other results . Note that the sample space depends on the observations (compare Examples 1 and 2 or Examples 3 and 4). S is a discrete sample space if the number of outcomes is countable (maps to the positive integers). S is a continuous sample space if S is not countable. An event specifies certain conditions for an outcome. It can be represented by a subset , A, of the sample space S. An event A occurs if the outcome is a member of A. The certain event , S , consists of all outcomes. The null event, φ, contains no outcomes. An elementary event contains only one outcome. Usually interested in events, rather than individual outcomes
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Elec210 Lecture 2 4 Discrete Sample Spaces Experiment 1 : Select a ball from an urn containing balls labeled 1 to 50. Note the number on the ball. S 1 = {1, 2, 3, 4, ……,49, 50} A 1 = “An even numbered ball is selected” = {2, 4, 6, …,48, 50} Experiment 2 : Select a ball from an urn containing balls labeled 1 to 50. Note the number in the tens place. S 2 = {0,1, 2, 3, 4,5} A 2 = “The number is more than 2” ={3, 4, 5} Experiment 3a : Toss a coin in the air and note which side faces up when it lands. The two sides of a coin are often called “heads” (H) and “tails” (T). S 3 b = {H,T} A 3 b = “The coin lands with heads up” = H Experiment 3b: Toss a coin three times. Note the sequence of heads/tails. S 3 b = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT} A 3 b = “The three tosses give the same outcome” = {HHH, TTT}
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Elec210 Lecture 2 5 Discrete Sample Spaces Experiment 4 : Toss a coin three times and note the number of heads. S 4 = {0, 1, 2, 3} A 4 = “The number of heads equals the number of tails” = φ Experiment 5 : A block of information is transmitted repeatedly over a noisy channel until an error-free block arrives at a receiver. Count the number of transmissions required. S 5 = {1, 2, 3, 4, ………. .} A 5 = “Less than 10 transmissions are required.” = {1, 2,. ., 8, 9} No outcome satisfies condition (IMPOSSIBLE EVENT)
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Elec210 Lecture 2 6 Continuous Sample Space Experiment 6 : Pick a number at random between zero and one. Experiment 7 : Measure the time between two typhoons. [] ( ] 1 , 5 . 0 } 1 5 . 0 : { 0.5" an greater th is number The " 1 , 0 } 1 0 : { 6 6 = < = = = = x x A x x S {} ) 2 , 0 [ 2 0 : elapse" months 2 than Less " ) , 0 [ } 0 : { 7 7 = < = = = = t t A t t S 0 1 6 S 6 A 0 2 7 S 7 A
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Probability Model Elec210 Lecture 2 7 Specifying Random Experiments Sample Space & Events Probability Model Practical Problem Axiomatic approach Probability model is generated using PROBABILITY THEORY -- General mathematical theory, independent of application
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Lecture2_Spring11 - Probability Model Axiomatic approach...

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