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 X, p as n → 0. This does not mean that X, will exactly equal to p, but rather the distribution of Xn is tightly concentrated around for large n. Suppose 0.1 < p < 0.9. Use Chebyshev's inequality to obtain a lower bound on P(p – 0.1 < X, < p+0.1). 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?
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 X, p as n → 0. This does not mean that X, will exactly equal to p, but rather the distribution of Xn is tightly concentrated around for large n. Suppose 0.1 < p < 0.9. Use Chebyshev's inequality to obtain a lower bound on P(p – 0.1 < X, < p+0.1). 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?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Problem 1P
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Transcribed Image Text: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 → 0. This
P
does not mean that X, will exactly equal to p, but rather the distribution of X, is tightly
concentrated around
for large n.
Suppose 0.1 < p < 0.9. Use Chebyshev's inequality to obtain a lower bound
on
Р(p - 0.1 < Х,
<р+0.1).
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:
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 < X, < 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 → x, and it only holds approximately for
finite n.
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