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# Empirical analyses

Empirical analyses question in R code way plus pdf formatPart 2: Central Limit Theorem The second part of the empirical analysis serve the purpose to get yml familiarizethe BLT. CLT characterize the following fact: Suppose {X..]- is a sequenCe of random variables draw from an underlying popu-lation space, then the sample mean of {Xe} de?ned as X, = i 3:, Xi, converge in distribution to Normal Distribution NEetFQJ, where ,e = E[X,-,}. 02 = [X] 1′!- {a} e Randomly draw l???UUDU samples from from N(2?,25}. plot the histogramof the stunpies and underlying population distribution together I What is the theoretical expectation for the population space? ~’varianee’? andStandard Deviation? I Let sample size 11 equal to ill and ll’l???ll respectively. We want to conductthe following experiment on both large sample cases (n = 100000] and sand]sample cases {n = 11)}. Repeat the following process for 2000 times: Randomly choose n samplesfrom the original l??????? sample space. [This is for speed consideration,you could draw n samples from N[2{l,25) directly each time, but that wouldcost a lot of time), compute the sample mean E, for each :1. As a result, we will have Z?’?? f1. and 20110 fll??l?? 0 Plot the histogram of sues sample mean Tm together with sugar, %). Whatis your conclusion from the graph. 0 Plot the histogram of sues sample mean 71mm: together with N??, mite}-1What is your conclusion from the graph. 0 Plot the histogram of ease normalized sample mean 72:" 51% together 1? ‘lf’ll m with NIH], 1}. What is your conclusion from the graph. 0 Plot the histogram of E??? normalized sample mean 3’23 Wtogether  ‘Ifl TUL-msr with NIH], 1}. 1What is your conclusion from the graph. I Change the parameters of the underlying distribution N?l}, 25} to what-ever you like, follow the same procedure again. Can you draw the sameconclusion? {b} Do the same thing as question (1} asked for Binomial Distribution EMU, 11.2] asthe underlying population distribution. {c} Do the same thing as question {1} asked for t-Distribution t{1?] as the underly-ing population distribution. 1When you change your own parameters as the lastquestion asked. make sure the degree of freedom be greater than 2, otherwise thesample mean may not converge in Normal Distribution, think about why?

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