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ECE313.Lecture05 - ECE 313 Probability with Engineering...

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The Axioms of Probability, Part III 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 5 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 33 Topics studied in previous class Sampling without replacement and its relation to random samples Sample spaces with countably infinite outcomes Axiom III for countably infinite disjoint sets Relative frequencies; many trials Uncountably infinite sample spaces Real numbers versus reality
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ECE 313 - Lecture 5 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 33 Measurements are rational A physical measurement made with an instrument yields a rational number At the microscopic level, most physical phenomena are discrete Any electrical charge is an integer multiple of the electrical charge of an electron Electrical current (charge/unit time) is a rational multiple of the quantity one electron charge/s
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ECE 313 - Lecture 5 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 33 But, the models are real numbers The real numbers in computer programs are actually rational numbers — irrational numbers cannot be represented exactly Then, why are physical parameters modeled as continuous variables? Easier to get (the “right”) answers di dt Calculus can be used: L + Ri = v assumes that i is a continuous function of t
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ECE 313 - Lecture 5 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 33 Measurements versus models All actual measurements result in rational numbers, but we model the measurement as being an arbitrary real number We all understand that V = 1.235 volts really means that V is some real number in the range 1.2345 ≤ V ≤ 1.2355 volts This modeling is useful and convenient in the physical sciences and engineering but causes difficulties in probability theory
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ECE 313 - Lecture 5 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 33 Real numbers in probability Uncountably infinite sample spaces are the real line (or intervals thereof ) Such spaces cause subtle mathematical difficulties in probability theory The resolution of these difficulties led to the development of the axiomatic theory of probability We will look only at the end results, not the gory details
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ECE 313 - Lecture 5 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 33 Relative frequencies converge to 0 The output of rand() is a good model for repeated trials for the experiment of picking a number at random in (0, 1) Every call to rand() returns a different number Relative frequency of a specific number in (0,1) converges to 0 Actually, the output of rand() is periodic (with long period,) and the numbers will repeat
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