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ECE313.Lecture40 - ECE 313 Probability with Engineering...

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Mean-Square Estimation 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 40 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Y is a RV with known pmf or pdf Problem: Predict (or estimate ) what value of Y will be observed on the next trial What value should we predict? What is a good prediction? We need to specify some criterion that determines what is a good/reasonable estimate Else any estimate is just as good as any other estimate Predicting the value of Y
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ECE 313 - Lecture 40 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 ê denotes our estimate of the value of Y ê is a number that we get to choose Minimum-probability-of-error criterion: choose α so as to minimize P{ Y ≠ ê} If Y is a discrete random variable, then P{ Y ≠ ê} = 1 – P{ Y = ê} = 1 – p Y (ê) Choose ê to be the location of the maximum of the pmf p Y (u) Our estimate is wrong with probability 1 – p Y (ê) Minimize probability of error – I
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ECE 313 - Lecture 40 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 If Y is a continuous random variable, then P{ Y ≠ ê} = 1 – P{ Y = ê} = 1 no matter what we number we choose as ê Alternative: choose ê to minimize P{| Y –ê| > ε } for some suitable (small) choice of ε P{| Y –ê| > ε } = 1 – P{| Y –ê| ≤ ε } and so we want to maximize P{| Y –ê| ≤ ε } P{| Y –ê| ≤ ε } = P{ê – ε Y ≤ ê + ε } = F Y (ê + ε ) – F Y (ê – ε ) Minimize probability of error – II
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ECE 313 - Lecture 40 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 P{| Y –ê| ≤ ε } = F Y (ê + ε ) – F Y (ê – ε ) has derivative f Y (ê + ε ) – f Y (ê – ε ) = 0 if ê is chosen such that f Y (ê + ε ) = f Y (ê – ε ) Graphically, find a horizontal “chord” of length 2 ε under the “peak” of the pdf: the midpoint of the chord is ê Minimize probability of error – III ê 2 ε
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ECE 313 - Lecture 40 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 In the limit as ε 0, ê approaches the location of the maximum value of the pdf For both continuous and discrete RVs, we get the location of the maximum of the pdf or the pmf The location of the maximum of the pdf or the pmf is called the mode of the pdf/pmf It is the value of Y that has the “maximum probability” of occuring Mode = most fashionable or most frequent What’s apple pie à la mode?
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ECE 313 - Lecture 40 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 If our estimate is ê, then we make an estimation error of Y – ê Cost of making this error is | Y – ê| Large estimation errors cost us more than small estimation errors
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