Question: A random variable is said to have the (standard) Cauchy distribution if its PDF is given by (5.2.6). This exercise uses computer simulations to demonstrate that a) samples from this distribution often have extreme outliers (a consequence of the heavy tails of the distribution), and (b) the sample mean is prone to the same type of outliers. (In fact, for any sample size, the sample mean has the standard Cauchy distribution, implying that the LLN and CLT do not apply for samples from a Cauchy distribution.)

(a) The R commands x=rcauchy(500); summary(x) generate a random sample of size 500 from the Cauchy distribution and display the sample's five number summary; see Section 1.7. Report the five number summary and the interquartile range, and comment on whether or not the smallest and largest order statistics are outliers. Repeat this 10 times.

(b) The R commands m=matrix(rcauchy(50000), nrow=500); xb=apply(m, 1, mean); summary(xb) generate the matrix m that has 500 rows, each of which is a sample of size n = 100 from the Cauchy distribution, compute the 500 sample means and store them in xb, and display the five number summary of xb. Repeat these commands 10 times, and report the 10 sets of five number summaries. Compare with the 10 sets of five number summaries from part (a), and comment on whether or not the distribution of the averages seems to be as prone to extreme outliers as that of the individual observations.