Mathematics for Computer Science Example: IQ MIT 6.042J/18.062J IQ measure was constructed so that Bounds on Deviation average IQ = 100. Markov Bound What fraction of the people can possibly have an IQ ≥ 300? …at most 1/3 markov.1 markov.2 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 IQ Higher than 300? IQ Higher than 300? At most 1/3 of people If more than 1/3 have have IQ ≥ 300 IQ ≥ 300, then avg > (1/3) · 300 = 100 ! Pr[IQ ≥ 300] ≤ E[IQ] —a contradiction 300 Albert R Meyer, May 10, 2013 markov.3 Albert R Meyer, May 10, 2013 markov.5 1

IQ Higher than x? IQ Higher than x? In general, Besides mean = 100, Pr[IQ ≥ x] ≤ 100 we used only one fact about the distribution of IQ: x IQ is always nonnegative markov.6 markov.7 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 Markov Bound (Restated) Markov Bound Let x = c·E[R]: If R is nonnegative, then Pr[R ≥ c µ ] ≤ 1 Pr[R ≥ x] ≤ E R c x Pr[R ≥ 3·expected] ≤ 1/3 for x > E[R] 0 Albert R Meyer, May 10, 2013 markov.8 Albert R Meyer, May 10, 2013 markov.9 2

IQ ≥ 300, again Markov Bound Suppose we are given that •Weak IQ is always ≥ 50? Get a better bound using •Obvious (IQ – 50) •Useful anyway since this is now ≥ 0. markov.10 markov.11 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 IQ ≥ 300, again Improved Markov Bound Pr[IQ ≥ 300] = Better bound from ≥ 300 Pr[IQ -50 -50] Markov by shifting R ≤ 100 -50 300 -50 = 1 to have 0 as minimum 5 Albert R Meyer, May 10, 2013 markov.12 Albert R Meyer, May 10, 2013 markov.13 3

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