STAT312 Lecture : Central Limit Theorem.pdf
5 Apr 2014
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Proof of clt: we make the additional assump- have an m. g. f. We are to show that the m. g. f. of tends to that of a. )# = y=1: r. v. , i. e. that tends to. We use the following notation: ( ) = ( ( )) as. 000( ) has a nite limit as log ( If = (constant) then all occurrences of replaced by. 6. ) can be (in this case (1) was proven in lab. Application: we often make inferences about a population mean using the -statistic where is as in the clt and is the sample standard deviation. If the data are normally dis- tributed then follows a student s t distribution. 1 degrees of freedom; it is well known that on this distribution is closely approximated by the is reasonably large. This latter fact holds even for non-normal parent distribu- tions: (0 1) when. 2 (since is a continuous function of: and so.
