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**Unformatted text preview: **P (40 < X < 60). Solution: 1. By Markovs inequality, P ( X > 75) 50 75 = 2 3 2. Write P (40 < X < 60) = 1-P ( | X-50 | 10). By Chebyshevs 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 Markovs Inequality; Chebyshevs Inequality The WLLN and the CLT Theorem (The Weak Law of Large Numbers) Let X 1 , X 2 , . . . be iid with nite mean , and set X n = 1 n n i =1 X i . Then, for any > , P ( | X n- | > ) , as n Theorem (The Central Limit Theorem) Let X 1 , X 2 , . . . be iid with mean and nite variave 2 , and set X n = 1 n n i =1 X i . Then X n N , 2 n X 1 + + X n N ( n , n 2 ) Michael Akritas The Law of Large Numbers and The Central Limit Theorem...

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