Lecture9 - Law of Large Numbers • An intuitive way to view the probability of a certain outcome is the frequency with which that outcome occurs

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Unformatted text preview: Law of Large Numbers 10/27/2005 • 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 Chebyshev Inequality Theorem. Let X be a discrete random variable with expected value μ = E ( X ) , and let > 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 Example • Let X by any random variable with E ( X ) = μ and V ( X ) = σ 2 . • Then, if = kσ , Chebyshev’s Inequality states that P ( | X- μ | ≥ kσ ) ≤ σ 2 k 2 σ 2 = 1 k 2 ....
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This note was uploaded on 07/16/2010 for the course MATH 20 taught by Professor Ionescu during the Fall '05 term at Dartmouth.

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Lecture9 - Law of Large Numbers • An intuitive way to view the probability of a certain outcome is the frequency with which that outcome occurs

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