# rec15 - d i.e for a given accuracy d and given conFdence p...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Recitation 15 April 20, 2006 1. Let X and Y be random variables, and let a and b be scalars; X takes nonnegative values. (a) Use the Markov inequality on the random variable e sY to show that sb M Y ( s ) , P ( Y b ) e for every s > 0, where M Y ( s ) is the transform of Y . 2. Joe wishes to estimate the true fraction f of smokers in a large population without asking each and every person. He plans to select n people at random and then employ the estimator F = S/n , where S denotes the number of people in a size- n sample who are smokers. Joe would like to sample the minimum number of people, but also guarantee an upper bound p on the probability that the estimator F differs from the true value f by a value greater than or equal to d i.e., for a given accuracy d and given confidence p , Joe wishes to select
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Unformatted text preview: d i.e., for a given accuracy d and given conFdence p , Joe wishes to select the minimum n such that P ( | F − f | ≥ d ) ≤ p . ±or p = 0 . 05 and a particular value of d , Joe uses the Chebyshev inequality to conclude that n must be at least 50,000. Determine the new minimum value for n if: (a) the value of d is reduced to half of its original value. (b) the probability p is reduced to half of its original value, or p = 0 . 025. 3. Let X 1 ,X 2 ,... be a sequence of independent random variables that are uniformly distributed between 0 and 1. ±or every n , we let Y n be the median of the values of X 1 ,X 2 ,... ,X 2 n +1 . [That is, we order X 1 ,... ,X 2 n +1 in increasing order and let Y n be the ( n + 1)st element in this ordered sequence.] Show that the sequence Y n converges to 1/2, in probability. Page 1 of 1...
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