ECE313.Lecture10

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

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Confdence 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
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 justiFed 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
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 confdence interval is its confdence level The narrower the confdence interval, the lower the confdence 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 Confdence intervals and levels How does one fnd a confdence interval? IF 5,023 Heads occurred on 10,000 tosses oF a coin, and we want a confdence interval oF length 0.1, where in the interval [0,1] should our confdence interval be? What is the confdence level associated with this confdence interval? How to fnd a confdence interval For a specifed confdence level?
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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## 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.Lecture10 - ECE 313 Probability with Engineering...

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