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R2 Let X ~ N(μ = 10.82). Following up on RI, we will be approximating μ2, which we can see should be 100. For now, let the sample size be n 3, Pick 3 random numbers from X, compute 72 X, and repeat the process a total of 50000 times. Plot a smooth version of the histogram of these 50000 values for X2: the plot(density(...). command in R will be useful. Now find the average of your 50000 values and make a vertical dotted line in R at this number (match the color of your curve). You have just made a rough picture of the density function for X (when n 3) and identified its (approximate) expected value. Now you should repeat this process for n = 6, n = 10 and n-50, Plot all four of these curves (use different colors) and vertical lines on the same set of axes choosing limits on the axes so the picture looks nice. Finally, put a dark solid vertical line at x-112-100. You should be able to see the asymptotic unbiasedness in the picture. Your answer is your code and a rough sketch of the plot you have created.

math   Statistics-And-Probability   
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vtion the Follo di bing is the R code (ail stoliments shor ting e folo tbing is the R co de those ommento) ent the mean to uc-10 #3ct the 3 tandald deviation na 2-0 # get the Band om seed set. Seed 1123) moke an empty plot to add the cuves taton pot (1, Egre i m c(o, 3), glim dlo, a 2), main bqyuote (the tot sibutiom of bar (x)12) 4 5et the ala tos tHo Hot (015 с-с (ed, blue, geen , magenta) 313e3 n<-c fox Set he gamPle -c (3.6, lo ,50)# do this Ros coch SamPle gite n i in L leng th ()H eoumbes of somples to be taken 3 500 dau a sample of 3420 n, ine maze h is into amatsix of n 0u and columns x- x ba><-apply (x, 2, mean) tt get the Squah?of me ana (density 1 x bas 2) , tye : 1, col-Colg[1]) # add the vetical ine abline (v:me freom noimol (16,2) H add a 3 an (xbar D, col colsC, 19) oline (V-m u2 / /wd: 2) add the legend iegerd top Dight, Paste (n,n, gep , col -a1y-) albThe distribution of X n-3 n-6 n=10 n=50 400 600 800 200 x2

toe lan See that as the Sample si2e inaeases, vestilal line moves close3 to the Solid line (o to 4, loo, shoaing he asgnp lobic un basednes 94 m oe lan also obsesue hat the vabiance of H educes oith the incease in sompe hat he density distai bution Finally e lon also see on a bell Sha pe as te 5am Ple Si?e incaeases in the asymp lo tic not mality The lade in text Fosmat is below #Set the mean mu )ó tt set the standa)d de uiation Sigina -6 Set Seed (13) # make an empty plot to odd the cu wes lalon im o0 nain bquote the distsibudin of bot() #Get He coloDS for the plot cols <-с (Dod., . ri blue, green, magenta get the Sample S12e5 n2-C3,6, 10,56

it do this fos cach Somple sine n o (i in 1. lengh (n) #gel tte numbe of Samples to be taken る1-50000 廿diaw a sample of sne nㆆ times, uat is nyy dsotus F3om nox mal (10, 2) tt make Hhis is into a matain of n 80u5 and catumns 丑gets 5ample means Xba appiy (x,1, mears) get He sa ua e of means H dsaw the density of Xbos 2 fines (density (ba)L) H add the ve tital lire abline (v: mcan ( bay 2), col: cols [),19.) # add a Solid line for mu Hl add the legend n sep legend ( top sight , paste (n

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