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Unformatted text preview: n observations is deﬁned by ¯ X n = ¯ X = X 1 + X 2 + ... + X n n , n = 1 , 2 ,... . Recall that E ( ¯ X n ) = μ, and Var ( ¯ X n ) = σ 2 n , n = 1 , 2 ,... . Conclusion : The sample mean becomes more and more concentrated around the (population) true mean μ . The Law of Large Numbers : The sample mean of a sequence of iid random variables with mean μ , converges to this true mean as the sample size n grows. ¯ X n → μ as n → ∞ . 1 . Markov’s inequality For any nonnegative random variable X and a number a > P ( X > a ) ≤ EX a . 2 . Chebyshev’s inequality For any random variable X with mean μ , and number h > P (  Xμ  > h ) ≤ Var( X ) h 2 . 3 . The Law of Large Numbers For any ± > P (± ± ¯ X nμ ± ± > ± ) → 0 as n → ∞ ....
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 '10
 SAMORODNITSKY
 Normal Distribution, Variance, Laplace, Probability theory, Characteristic function

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