The Cauchy distribution has the density function f (x) = 1

Chapter , Problem 38

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The Cauchy distribution has the density function

                                               \(f(x)=\frac{1}{\pi}\left(\frac{1}{1+x^2}\right), \quad-\infty<x<\infty\)

which is symmetric about zero. This distribution has very heavy tails, which cause the arithmetic mean to be a very poor estimate of location. Simulate the distribution of the arithmetic mean and of the median from a sample of size 25 from the Cauchy distribution by drawing 100 samples of size 25 and compare. From Example B in Section 3.6.1, if \(Z_1\) and \(Z_2\) are independent and N(0,1), then their quotient follows a Cauchy distribution. (This gives a simple way of generating Cauchy random variables.)