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sol13 - U.C Berkeley – CS70 Discrete Mathematics and...

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Unformatted text preview: U.C. Berkeley – CS70: Discrete Mathematics and Probability Problem Set 13 Lecturers: Umesh Vazirani & Christos H. Papadimitriou Problem Set 13 Solutions 1. We defined the probability distribution of the power law as: Pr[X=x] = C x –a . To find mean, we take ∑ ∑ ∞ < < ∞ < < +-- = × x x a a Cx Cx x 1 . We know that this series does not converge for a ≤ 2 and does converge for a > 2 thereby yielding a finite mean, μ . To find the variance, we compute the second moment ( 29 ∑ ∑ ∞ < < +- ∞ < <- = x a x a Cx Cx x 2 2 . Similarly, this series does not converge for a ≤ 3 and does converge for all a > 3. For a ≤ 2, the expectation diverges, so the variance is undefined and can be taken to be infinite. 2. a. Airline B has a better chance at LA (94% vs. 89%) and Chicago (76% vs. 70%) b. Airline A has a better chance (84% vs. 79%) c. This paradox is known as Simpson’s Paradox. It is actually not a paradox but is easily found from a mathematical standpoint because the probability does not take into account the size of the...
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sol13 - U.C Berkeley – CS70 Discrete Mathematics and...

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