ECE313.Lecture04

ECE313.Lecture04 - ECE 313 Probability with Engineering...

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The Axioms of Probability, Part II Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications

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ECE 313 - Lecture 4 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 42 Review of Lecture #3 We explored the consequences of the axioms of probability We discussed the notion of a partition and how partitions can be used to calculate the probability of an event We discussed binomial coefficients We looked at a simple combinatorial problem
ECE 313 - Lecture 4 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 42 Drawing a random sample Example: An urn contains 6 identical red balls R 1 , R 2 , R 3 , R 4 , R 5 , R 6 and 3 identical green balls G 1 , G 2 , G 3 . A trial of the experiment consists of simultaneously drawing two balls at random from the urn The outcomes of this experiment are subsets of size 2 of the form {R 1 ,R 5 } or {R 4 ,G 1 } or {G 2 ,G 3 }

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ECE 313 - Lecture 4 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 42 How many subsets of each kind? There are = 36 subsets of size 2 from a set of 9 balls, and the collection of these subsets is the sample space = 15 outcomes consist of two red balls = 3 outcomes have two green balls =18 outcomes have 1 red, 1 green 9 2 6 2 3 2 6 1 3 1
ECE 313 - Lecture 4 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 42 More probabilities A = {R 2 is in the sample drawn} B = {G 2 is in the sample drawn} What is P(A B) ? P(A) = 8/36 P(B) = 8/36 AB = {outcome = {R 2 , G 2 }} is a singleton event, and hence P(AB) = 1/36 P(A B) = P(A) + P(B) – P(AB) = 15/36 Exercise: What is P(A c B c )?

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ECE 313 - Lecture 4 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 42 Sampling without replacement A sample of size k from a set of size n is a subset of size k {R 2 , G 2 } is the same as {G 2 , R 2 } Instead of obtaining the k elements of the subset simultaneously, we could draw them out one at a time, each draw being carried out without replacing the previously drawn elements This is sampling without replacement
ECE 313 - Lecture 4 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 42 Sampling without replacement If a previously drawn element were to be put back, then it could be drawn again This is not what we want! Sampling without replacement results in

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This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

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ECE313.Lecture04 - ECE 313 Probability with Engineering...

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