MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Question 4 Consider flipping a (biased) coin for which the probability of head is p. The fraction of heads after n independent tosses is Xn. Law of large numbers imply that Xn → p as n → o. This does not mean that Xn will exactly equal to p, but rather the distribution of X is tightly concentrated around p for large n. (a) Suppose 0.1 < p< 0.9. Use Chebyshev's inequality to obtain a lower bound on P(p- 0.1 < X, <p+0.1). (b) Suppose p 0.6. Using the above lower bound derived using Chebyshev's inequality, how large should n be so that P(0.5 < Xn < 0.7) > 0.95? Note: The purpose of this question is to enhance your understanding of Chebyshev's inequality. In practice, the bound provided by Chebyshev's inequality is usually very "loose", in the sense that n actually only needs to be much smaller for P(0.5 < Xn < 0.7) > 0.95 to hold. A tighter bound for the setting we considered can be obtained by Hoeffding's inequality (Example 6.15 in Wasserman) or Central Limit Theorem (we will study this in class). Note that the bounds from Chebyshev's inequality or Hoeffding's inequality are finite sample in that they hold for any finite n. On the contrary, the bound from Central Limit Theorem is asymptotic, in that it is a statement about n o, and it only holds approximately for finite n.
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Transcribed Image Text:Question 4 Consider flipping a (biased) coin for which the probability of head is p. The fraction of heads after n independent tosses is Xn. Law of large numbers imply that Xn → p as n → o. This does not mean that Xn will exactly equal to p, but rather the distribution of X is tightly concentrated around p for large n. (a) Suppose 0.1 < p< 0.9. Use Chebyshev's inequality to obtain a lower bound on P(p- 0.1 < X, <p+0.1). (b) Suppose p 0.6. Using the above lower bound derived using Chebyshev's inequality, how large should n be so that P(0.5 < Xn < 0.7) > 0.95? Note: The purpose of this question is to enhance your understanding of Chebyshev's inequality. In practice, the bound provided by Chebyshev's inequality is usually very "loose", in the sense that n actually only needs to be much smaller for P(0.5 < Xn < 0.7) > 0.95 to hold. A tighter bound for the setting we considered can be obtained by Hoeffding's inequality (Example 6.15 in Wasserman) or Central Limit Theorem (we will study this in class). Note that the bounds from Chebyshev's inequality or Hoeffding's inequality are finite sample in that they hold for any finite n. On the contrary, the bound from Central Limit Theorem is asymptotic, in that it is a statement about n o, and it only holds approximately for finite n.
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