MIT6_042JS10_lec37

MIT6_042JS10_lec37 - Mathematics for Computer Science...

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1 lec 13F.1 Albert R Meyer, May 7, 2010 Deviation from the Mean Mathematics for Computer Science MIT 6.042J/18.062J lec 13F.15 Albert R Meyer, May 7, 2010 Example: IQ IQ measure was constructed so that average IQ = 100 . What fraction of the people can possibly have an IQ ! 300 ? lec 13F.16 Albert R Meyer, May 7, 2010 IQ Higher than 300? Fraction f with IQ ! 300 adds ! 300f to average, so 100 = avg IQ ! 300f: f " 100/300 = 1/3 lec 13F.17 Albert R Meyer, May 7, 2010 At most 1/3 of people have IQ ! 300 IQ Higher than 300? lec 13F.19 Albert R Meyer, May 7, 2010 IQ Higher than x ? Besides mean = 100, we used only one fact about the distribution of IQ: IQ is always nonnegative lec 13F.20 Albert R Meyer, May 7, 2010 Pr{ R ! x } " ER # $ % & x Markov Bound If R is nonnegative , then for x ! E[R]
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2 lec 13F.22 Albert R Meyer, May 7, 2010 ! Weak ! Obvious ! Useful anyway Markov Bound lec 13F.24 Albert R Meyer, May 7, 2010 Suppose we are given that IQ is always ! 50 ? Get a better bound using ( IQ 50) since this is now ! 0. IQ ! 300, again lec 13F.25 Albert R Meyer, May 7, 2010 f contributes ( 300-50) f to the average of ( IQ - 50) , so 50 = E[ IQ - 50] ! 250 f f " 50/250 = 1/5 IQ ! 300, again Better bound from Markov by shifting R to have 0 as minimum lec 13F.26 Albert R Meyer, May 7, 2010 Pr{ |R±²| ! x } = Pr{ (R±²) 2 ! x 2 } by Markov: Improving the Markov Bound variance of R lec 13F.28 Albert R Meyer,
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MIT6_042JS10_lec37 - Mathematics for Computer Science...

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