Cauchy distribution

The Cauchy distribution is a real, continuous, univariate Probability distribution. For a Random variable $X$ , the Probability density function is

f_{X}(x)=\frac{1}{\pi} \frac{1}{1+x^{2}}

It has no parameters.

The shape is similar to that of the Gaussian distribution, but the Cauchy distribution has heavier tails and is considerably less well-behaved. In fact, it is often studied specifically because of its difficult properties as a pathological case of probability distribution.

Moments#

The moment-generating function does not exist. The characteristic function does and should be used in its place. All function moments either diverge or are undefined.

The expected value is

\text{E}[X]=\int_{-\infty}^{\infty} \frac{1}{\pi} \frac{x}{1+x^{2}} \ dx =\frac{1}{\pi} \frac{1}{2}\ln(1+x^{2})|_{\infty}^{\infty}=\text{undefined}

Thus, the Cauchy distribution has no expected value. The variance is

\text{var}(X)=\int_{-\infty}^{\infty} \frac{1}{\pi} \frac{x^{2}}{1+x^{2}} \ dx \to \infty

which tends to infinity.¹ Notably, the lack of a well-defined expectation and variance imply that Chebyshev's inequality does not hold and neither does the central limit theorem.

Properties#

It is symmetrical around $x=0$ .
The sum of Cauchy-distributed iid variables is itself a Cauchy distribution.

Relation to other distributions#

The ratio $Y=\frac{X_{1}}{X_{2}}$ of two standard-normal iid variables $X_{1},X_{2}\sim \mathcal{N}(0,1)$ is a Cauchy distribution.
A Student's t distribution with one degree of freedom is a Cauchy distribution.
The Breit-Wigner distribution used in particle physics is a form of the Cauchy distribution.

This is somewhat improper notation. Formally, the integral should go from $-\infty$ to $a$ , then in the limit $a\to \infty$ the integral goes to infinity. ↩

Moments#

Properties#

Relation to other distributions#

Footnotes#