Bounds on Deviation average IQ = 100. Markov Bound What fraction - - PowerPoint PPT Presentation

markov.1 Albert R Meyer, May 10, 2013

Bounds on Deviation

Markov Bound

Mathematics for Computer Science

MIT 6.042J/18.062J

markov.2 Albert R Meyer, May 10, 2013

Example: IQ IQ measure was constructed so that

average IQ = 100.

What fraction of the people can possibly have an IQ ≥ 300?

…at most 1/3

markov.3 Albert R Meyer, May 10, 2013

IQ Higher than 300?

If more than 1/3 have IQ ≥ 300, then avg > (1/3)·300 = 100 ! —a contradiction

markov.5 Albert R Meyer, May 10, 2013

At most 1/3 of people have IQ ≥ 300

Pr[IQ ≥ 300] ≤ E[IQ] 300

IQ Higher than 300?

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markov.6 Albert R Meyer, May 10, 2013

In general,

IQ Higher than x?

Pr[IQ ≥ x] ≤ 100 x

markov.7 Albert R Meyer, May 10, 2013

IQ Higher than x?

Besides mean = 100, we used only one fact about the distribution of IQ:

IQ is always nonnegative

markov.8 Albert R Meyer, May 10, 2013

Pr[R ≥ x] ≤ E R     x

Markov Bound

If R is nonnegative, then

for x > E[R]

markov.9 Albert R Meyer, May 10, 2013

Let x = c·E[R]: Pr[R ≥ 3·expected] ≤ 1/3

Markov Bound (Restated)

Pr[R ≥ cµ] ≤ 1 c

2

markov.10 Albert R Meyer, May 10, 2013

• Weak
• Obvious
• Useful anyway

Markov Bound

markov.11 Albert R Meyer, May 10, 2013

Suppose we are given that IQ is always ≥ 50? Get a better bound using (IQ – 50) since this is now ≥ 0.

IQ ≥ 300, again

markov.12 Albert R Meyer, May 10, 2013

Pr[IQ -50 ≥ 300

• 50]

IQ ≥ 300, again

≤ 100 -50 300 -50 = 1 5 Pr[IQ ≥ 300] =

markov.13 Albert R Meyer, May 10, 2013

Improved Markov Bound

Better bound from Markov by shifting R to have 0 as minimum

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6.042J / 18.062J Mathematics for Computer Science

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