LLN_CLT

# LLN_CLT - P(40< X< 60 Solution 1 By Markov’s...

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Outline Markov’s Inequality; Chebyshev’s Inequality The WLLN and the CLT The Law of Large Numbers and The Central Limit Theorem Michael Akritas Michael Akritas The Law of Large Numbers and The Central Limit Theorem

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Outline Markov’s Inequality; Chebyshev’s Inequality The WLLN and the CLT Markov’s Inequality; Chebyshev’s Inequality The WLLN and the CLT Michael Akritas The Law of Large Numbers and The Central Limit Theorem
Outline Markov’s Inequality; Chebyshev’s Inequality The WLLN and the CLT Proposition (Markov’s Inequality) If X 0 , i.e. X is a nonnegative r.v., the P ( X > a ) E ( X ) a holds for any a > 0 Proposition (Chebyshev’s Inequality) For any random variable X with σ 2 X < , and any a > 0 , we have P ( | X - μ X | > a ) σ 2 X a 2 Michael Akritas The Law of Large Numbers and The Central Limit Theorem

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Outline Markov’s Inequality; Chebyshev’s Inequality The WLLN and the CLT Example Suppose the number X of items produced by a factory in a week is a random variable with mean 50, and variance 25. 1. Find an upper bound for P ( X 75). 2. Find a lower bound for
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Unformatted text preview: P (40 < X < 60). Solution: 1. By Markov’s inequality, P ( X > 75) ≤ 50 75 = 2 3 2. Write P (40 < X < 60) = 1-P ( | X-50 | ≥ 10). By Chebyshev’s inequality, P ( | X-50 | ≥ 10) ≤ 25 / 100 = 0 . 25. Hence, P (40 < X < 60) = 1-P ( | X-50 | ≥ 10) ≥ 1-. 25 = 0 . 75 . Michael Akritas The Law of Large Numbers and The Central Limit Theorem Outline Markov’s Inequality; Chebyshev’s Inequality The WLLN and the CLT Theorem (The Weak Law of Large Numbers) Let X 1 , X 2 , . . . be iid with ﬁnite mean μ , and set X n = 1 n ∑ n i =1 X i . Then, for any ± > , P ( | X n-μ | > ± ) → , as n → ∞ Theorem (The Central Limit Theorem) Let X 1 , X 2 , . . . be iid with mean μ and ﬁnite variave σ 2 , and set X n = 1 n ∑ n i =1 X i . Then X n ˙ ∼ N ± μ, σ 2 n ² X 1 + ··· + X n ˙ ∼ N ( n μ, n σ 2 ) Michael Akritas The Law of Large Numbers and The Central Limit Theorem...
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LLN_CLT - P(40< X< 60 Solution 1 By Markov’s...

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