A random variable is said to have the standard Cauchy distri

A random variable is said to have the (standard) Cauchy distribution if its PDF is given by This problem uses computer simulations to demonstrate dial 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. Below is a graph of the pdf The R commands x-raunchy(500) summary(x) generate a random sample of si/c 500 from the Cauchy distribution and display the sample\'s five number summary; Report the five number summary and the interquamle range, and comment on whether or not the smallest and largest numbers generated from this sample of 500 are outliers. Repeat this 10 times. The R commands m matrix(raunchy(50000). nrow=500); xb-apply( m. 1 .mean) summary (xb) generate the matrix m that has 500 row s, 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 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 more prone to extreme outliers as that of the individual observations. Why docs this happen? and docs the LLN and C\'LT apply for samples from a Cauchy distribution?

Solution

a) the 10 samples summary is given

1) Min. 1st Qu. Median Mean 3rd Qu. Max.
-121.8000 -1.0420 0.0206 2.7320 1.1270 506.9000

2) Min. 1st Qu. Median Mean 3rd Qu. Max.
-56.38000 -0.86000 -0.01422 0.90530 0.98510 144.10000
3) Min. 1st Qu. Median Mean 3rd Qu. Max.
-176.20000 -0.99880 0.04283 0.25330 0.89900 239.90000
4) Min. 1st Qu. Median Mean 3rd Qu. Max.
-7772.000 -1.126 -0.191 -17.480 0.893 74.730
5) Min. 1st Qu. Median Mean 3rd Qu. Max.
-71.30000 -1.02900 0.03723 0.75600 0.94310 177.10000
6) Min. 1st Qu. Median Mean 3rd Qu. Max.
-253.8000 -1.2130 0.0221 2.4030 1.0330 1660.0000
7) Min. 1st Qu. Median Mean 3rd Qu. Max.
-245.3000 -0.8437 -0.0113 4.1900 0.9736 2020.0000
8) Min. 1st Qu. Median Mean 3rd Qu. Max.
-212.3000 -1.1450 -0.0196 0.7689 0.9351 407.8000

9) Min. 1st Qu. Median Mean 3rd Qu. Max.
-488.100 -0.749 0.181 31.530 1.295 11400.000
10) Min. 1st Qu. Median Mean 3rd Qu. Max.
-95.24000 -0.90970 -0.00979 0.53470 0.96900 260.10000