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STAT1301 (Ch4)

STAT1301 (Ch4) - Stat 1301B Probability Statistics I Fall...

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Stat 1301B Probability & Statistics I Fall 2009-2010 Chapter IV Limit Theorems The probabilistic behaviour of the sample mean when the sample size n is large (say, tends to infinity) is called the limiting distribution of the sample mean. Law of large number ( LLN ) and the central limit theorem ( CLT ) are two of the most important theorems in statistics concerning the limiting distribution of the sample mean. These two theorems suggest the “nice” properties of the sample mean and justify its advantages. Before proceeding, we need to define what ‘convergence’ means in the context of random variables. § 4.1 Modes of Convergence Let be a sequence of random variables (not necessarily independent), X be another random variable. Let ,... , 2 1 X X ( ) x F n X ( ) x F X be the distribution function of , n X be the distribution function of X . Converges in Distribution / Converges in Law / Weak Convergence n X is said to converge in distribution to X if ( ) ( ) x F x F X X n n = lim for all points x at which is continuous. It is denoted as . ( ) x F X X X L n ⎯→ Example 4.1 Let . Define as the maximum of . Then the distribution function of is given by ( 1 , 0 ~ ,... , 2 1 U U U iid ) n X n U U U ,..., , 2 1 n X ( ) 0 = x F n X for 0 x ; ( ) 1 = x F n X for 1 x ; ( ) ( ) ( ) x U x U x U P x X P x F n n X n = = ,..., , 2 1 ( ) ( ) ( ) x U P x U P x U P n = L 2 1 n x = , for 1 0 < < x . P.152

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