Standard Deviation of Cauchy Distribution

Date: 06/12/2005 at 19:14:01
From: Helen
Subject: standard deviation of Cauchy distribution

If the variance of a probability distribution for a continuous random 
variable with a mean of zero can be found by integrating (x-squared 
times f(x)) between the end bounds of distribution, and the indefinite 
integral is easily obtained, but leads to an infinite value when the 
definite integral is calculated, how do you work out the standard 
deviation of the Cauchy distribution?

I'm having difficulty with the fact that the continuous random 
variable, X, is defined for all reals.

I can't show my work because I can't paste MathType, but I get  
(1/pi)[x - arctanx] between -infinity and infinity.

Date: 06/13/2005 at 11:26:50
From: Doctor George
Subject: Re: standard deviation of Cauchy distribution

Hi Helen,

Thanks for writing to Doctor Math.

The answer is that the Cauchy distribution does not have a standard
deviation.  That is a disturbing result, but it is true.

The width is characterized by its "full width at half maximum."  Take
a look at this link:

  Cauchy Distribution
    http://mathworld.wolfram.com/CauchyDistribution.html 

Strictly speaking, the Cauchy distribution does not have a mean 
either, but it does have a median.

What this means on a practical level is that the Central Limit Theorem
does not apply to the Cauchy distribution.  The distribution of the
sample mean also has a Cauchy distribution, and it does not approach
the normal distribution as the number of samples increases.

Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/