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Math Help - Chebyshev and Central Limit Theorem

  1. #1
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    Chebyshev and Central Limit Theorem

    Let S be the number of heads in 1,000,000 tosses of a fair coin. How does on use (a) Chebyshev's inequality and (b) the Central Limit Theorem to estimate the probability that S lies between 499,500 and 500,500?

    Thanks so much!!!
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  2. #2
    MHF Contributor matheagle's Avatar
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    n=1,000,000 and p=.5 approximating this binomial with a normal (CLT) you need

    \mu=np and \sigma^2=npq

    Since you want the probability of an event that has \mu at its center this is straightforward.

    P(499,500 < X_B < 500,500)\approx P\biggl({499,500-\mu\over\sigma} < Z < {500,500-\mu\over\sigma}\biggr)

    As for cheby's figure out your k at http://en.wikipedia.org/wiki/Chebyshev's_inequality
    look under Probabilistic statement
    Last edited by matheagle; August 2nd 2009 at 08:37 PM.
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  3. #3
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    I'm just confused because for this problem doesn't k=1 for Chebyshev so that 1/k^2 = 1 which means that there is a 0 probability that S lies between 499,500 and 500,500?
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