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# chap5 - Chapter 5 Limit Theorems Let X1 Xn be independent...

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Chapter 5 Limit Theorems Let X 1 , . . . , X n be independent and identically distributed random variables with mean μ and variance σ 2 . If M n = 1 n ( X 1 + . . . + X n ) then E [ M n ] = μ and var( M n ) = σ 2 n . As n + , var( M n ) 0 and “ M n approaches μ ” (in some sense). This is a “first order approximation” of M n . The Central Limit Theorem gives a “second order approximation” of M n : M n - μ σ/ n converges (in some sense) to a standard normal random variable. 1 Markov and Chebyshev’s Inequalities If X is a nonnegative random variable, then P [ X a ] E [ X ] a , a > 0 . 1

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2 MTH2222 – Mathematics of Uncertainty Let X be exponential λ . Compare the Markov bound to the exact probability P [ X a ]. If X is a random variable with mean μ and variance σ 2 , then P [ | X - μ | ≥ c ] σ 2 c 2 , c > 0 . In particular P [ | X - μ | ≥ ] 1 k 2 , k > 0 . 2 The Weak Law of Large Numbers Let X 1 , . . . , X n be independent and identically distributed random variables with mean μ and variance σ 2 . For every ε > 0, we have P [ |
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chap5 - Chapter 5 Limit Theorems Let X1 Xn be independent...

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