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Lecture9 - Law of Large Numbers An intuitive way to view...

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Law of Large Numbers 10/27/2005
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An intuitive way to view the probability of a certain outcome is the frequency with which that outcome occurs in the long run. We defined probability mathematically as a value of a distribution function for the random variable representing the experiment. The Law of Large Numbers shows that this model is consistent with the frequency interpretation of probability. 1
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Chebyshev Inequality Theorem. Let X be a discrete random variable with expected value μ = E ( X ) , and let > 0 be any positive real number. Then P ( | X - μ | ≥ ) V ( X ) 2 . Proof. Let m ( x ) denote the distribution function of X . P ( | X - μ | ≥ ) = X | x - μ |≥ m ( x ) . P ( | X - μ | ≥ ) = X | x - μ |≥ m ( x ) . 2 2
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Example Let X by any random variable with E ( X ) = μ and V ( X ) = σ 2 . Then, if = , Chebyshev’s Inequality states that P ( | X - μ | ≥ ) σ 2 k 2 σ 2 = 1 k 2 . Thus, for any random variable, the probability of a deviation from the mean of more than k standard deviations is 1 /k 2 .
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