Suppose a random variable Y is defined as
where
The Cauchy distribution is specified with two parameters:
a and b. We denote it
The Cauchy distribution has probability density function (PDF)
This is graphed in Exhibit 1.
The PDF is more peaked in the middle and has fatter tails than a normal distribution. For this reason, we might want to call the distribution leptokurtic. That would technically not be correct because the distribution's kurtosis is undefined. Actually, its mean, standard deviation and skewness are also not defined. As Exhibit 1 indicates, the distribution's median and mode both occur at a. The parameter b is a dispersion parameter, playing a role similar to that of a standard deviation. The cumulative distribution function is
Any Cauchy random variable X ~
The Cauchy distribution is a
stable Paretian
distribution, so a sum of Cauchy random variables is itself Cauchy. More
precisely, consider n independent random variables
The Cauchy distribution has characteristic function
The standard Cauchy distribution is a special case of the student t distribution with one degree of freedom.
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and
are independent 
. The
random variable Y defined in [
,
and let Y equal their sum. Then
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