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 f(x) = 1/pi 1/1 + x^2, -infinity

Solution

a)

y=list()
> for(i in 1:10){
+ x=rcauchy(500)
+ y[[i]]=summary(x)
+ }
> y
[[1]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-149.700 -1.129 -0.158 20.000 0.831 10040.000

[[2]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-109.3000 -0.7500 0.1714 1.0720 1.2870 211.6000

[[3]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-99.9600 -0.8090 0.1533 1.6270 1.3710 260.4000

[[4]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-180.30000 -1.08200 0.00473 0.06115 1.25400 110.60000

[[5]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-781.8000 -1.0300 -0.0142 -1.8890 0.9732 83.7100

[[6]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-688.7000 -0.9876 -0.0452 -1.6580 0.9300 153.1000

[[7]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-38.96000 -0.91290 0.09453 0.07371 1.04700 87.15000

[[8]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-473.2000 -0.7780 0.0624 0.2948 1.0720 216.7000

[[9]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-106.60000 -0.96270 -0.02837 -0.21190 0.85390 86.25000

[[10]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-52.47000 -1.18600 -0.04586 0.39870 0.78900 166.20000

Yes, the largest and smallest number generated from the sample 500 are outliers for each time.

b)

> z=list()
> for(i in 1:10){
+ m=matrix(rcauchy(50000), nrow=500)
+ xb=apply(m,1,mean)
+ z[[i]]=summary(xb)
+ }
> z
[[1]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-287.10000 -0.90430 0.09803 -0.98110 1.08000 133.90000

[[2]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-332.4000 -1.0660 0.0083 -0.4087 0.9121 185.0000

[[3]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-279.9000 -1.2160 -0.0683 -0.5834 0.9216 421.7000

[[4]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-98.2600 -1.1380 -0.0708 1.7990 0.8974 642.7000

[[5]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-423.500 -0.935 -0.025 16.640 0.866 8584.000

[[6]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-72.4300 -1.1030 -0.0146 2.6100 0.9645 1001.0000

[[7]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-481.5000 -0.8742 0.0212 0.2480 1.0460 220.8000

[[8]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-313.8000 -0.9905 -0.0387 3.0460 0.9359 1762.0000

[[9]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-150.10000 -0.97250 -0.01757 0.48870 0.94460 257.90000

[[10]]
Min. 1st Qu. Median Mean 3rd Qu. Max.
-57.33000 -0.89850 0.04766 0.69130 0.94000 105.80000

Yes the average seems to be more prone to extreme outliers as that of the individual observations.