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ECE313.Lecture10 - ECE 313 Probability with Engineering...

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Confidence Intervals 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 10 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Review I Y denotes the number of occurrences of an event A of probability p on n trials Y is a binomial random variable with parameters (n, p). It has mean E[ Y ] = np, variance np(1–p), and mode (n+1)p Problem: Y had value k on a trial of the (compound) experiment. Estimate the unknown value of p from this datum The relative frequency estimate of p is k/n
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ECE 313 - Lecture 10 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 Review II The relative frequency estimate can be justified via maximum-likelihood principle Maximum-likelihood (ML) principle: the estimate of the value of an unknown parameter is the number that maximizes the likelihood of the observation If an event A occurred k times on n trials, the ML estimate of P(A) is k/n, which is the same as the relative frequency estimate
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ECE 313 - Lecture 10 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 Review III The value of the unknown parameter p is assumed to be any number in [0, 1] It is only in fortuituous circumstances that a point estimate such as the ML estimate will be the exact value of p More often than not, a point estimate will be close , but not exactly right Attempting to get a more precise estimate only reduces the chances of accuracy
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ECE 313 - Lecture 10 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 Review IV Point estimates (e.g. the ML estimate k/n) versus interval estimates such as “0.5013 < p < 0.5033” or “k/n ± 3%” Associated with a confidence interval is its confidence level The narrower the confidence interval, the lower the confidence level
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ECE 313 - Lecture 10 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 Confidence intervals and levels How does one find a confidence interval? If 5,023 Heads occurred on 10,000 tosses of a coin, and we want a confidence interval of length 0.1, where in the interval [0,1] should our confidence interval be? What is the confidence level associated with this confidence interval? How to find a confidence interval for a specified confidence level?
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ECE 313 - Lecture 10 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 Back to probability … for a while The variance of a random variable is a measure of the spread of the probability masses about the mean µ The larger the variance, the wider the dispersion of the masses away from µ How much probability mass lies “far away” from µ?
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