Cauchy Distribution

Explained:

Suppose a random variable Y is defined as

[1]

where and are independent standard normal random variables. How is Y distributed? The answer is that it has a Cauchy distribution.

The Cauchy distribution is specified with two parameters: a and b. We denote it . The random variable Y defined in [1] above actually has a C(0,1) distribution, which is called the standard Cauchy distribution.

The Cauchy distribution has probability density function (PDF)

[2]

This is graphed in Exhibit 1.

C(a,b) Probability Density Function
Exhibit 1

The C(a,b) distribution has a symmetric "bell shaped" probability density function, but it is more peaked at the center and has fatter tails than a normal distribution.

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

[3]

Any Cauchy random variable X ~ can be expressed in terms of a standard Cauchy variable Z ~ C(0,1) as

X = bZ + a

[4]

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 , and let Y equal their sum. Then

[5]

The Cauchy distribution has characteristic function

[6]

The standard Cauchy distribution is a special case of the student t distribution with one degree of freedom.