Standard Deviation of Cauchy DistributionDate: 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/ |
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