Stat 1301B Probability & Statistics I Fall 2009-2010Chapter IV Limit TheoremsThe probabilistic behaviour of the sample mean when the sample size nis large (say, tends to infinity) is called the limiting distributionof 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 ConvergenceLet be a sequence of random variables (not necessarily independent), Xbe another random variable. Let ,...,21XX( )xFnX( )xFXbe the distribution function of , nXbe the distribution function of X. Converges in Distribution / Converges in Law / Weak Convergence nXis said to converge in distribution to Xif ( )( )xFxFXXnn=∞→limfor all points xat which is continuous. It is denoted as . ( )xFXXXLn⎯→⎯Example 4.1Let . Define as the maximum of . Then the distribution function of is given by (1,0~,...,21UUUiid)nXnUUU,...,,21nX( )0=xFnXfor 0≤x; ( )1=xFnXfor 1≥x; ( )()()xUxUxUPxXPxFnnXn≤≤≤=≤=,...,,21() ()()xUPxUPxUPn≤≤≤=L21nx=, for 10<<x. P.152
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