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MIT6_042JS10_lec37_sol

MIT6_042JS10_lec37_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science May 7 Prof. Albert R. Meyer revised May 5, 2010, 857 minutes Solutions to In-Class Problems Week 13, Fri. Problem 1. A herd of cows is stricken by an outbreak of cold cow disease . The disease lowers the normal body temperature of a cow, and a cow will die if its temperature goes below 90 degrees F. The disease epidemic is so intense that it lowered the average temperature of the herd to 85 degrees. Body temperatures as low as 70 degrees, but no lower , were actually found in the herd. (a) Prove that at most 3/4 of the cows could have survived. Hint: Let T be the temperature of a random cow. Make use of Markov’s bound. Solution. Let T be the temperature of a random cow. Then the fraction of cows that survive is the probability that T 90 , and E [ T ] is the average temperature of the herd. Applying Markov’s Bound to T : Pr { T 90 } = E [ T ] = 85 = 17 . 90 90 18 But 17 / 18 > 3 / 4 , so this bound is not good enough. Instead, we apply Markov’s Bound to T 70 : Pr { T 90 } = Pr { T 70 20 } ≤ E [ T 20 70] = (85 70) / 20 = 3 / 4 .
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