The Cauchy distribution, named after Augustin Cauchy, is usually a continuous probability submission.
The simplest Cauchy distribution is known as the standard Cauchy distribution. It is the distribution of an random variable, this is the ratio of a couple of independent standard normal variables. Mean and its variance are undefined. The Cauchy distribution does not have any moment generating perform. In this distribution Mode and Average are zero. A widely used measure of the width could be the full width at half maximum (FWHM), that's equal to 3.

A random variable X is said to have a standard cauchy distribution if its pdf is given by:

f(x) = $\frac{1}{\pi (1 + x^{2})}$ ; -$\infty$ < x < $\infty$

and X is said to be the standard cauchy variate.
If X is a standard cauchy variate then

$\phi_{X}$(t) = $\frac{1}{\pi}$ $\int_{-\infty}^{\infty}$ $\frac{e^{itx}}{1+x^{2}}$ $dx$

Now consider standard laplace distribution

f$_{1}$z =  $\frac{1}{2}$ e$^{-|z|}; -\infty < Z < \infty$

Then $\phi_{1}(t)$ = $\phi_{Z}(t)$ = $E (e^{itZ})$= $\frac{1}{1+t^{2}}$

Since $\phi_{1}(t)$ is absolutely integrable in (-$\infty, \infty$)

By inversion theorem we have

$\frac{1}{2}$ e$^{-|z|}$ = f$_{1}$(z)

= $\frac{1}{2\pi}$ $\int_{-\infty}^{\infty} e^{-itz}\phi_{1}$(t)dt

= $\frac{1}{2 \pi}$ $\int_{-\infty}^{\infty}$ $\frac{e^{itz}}{1 + t^{2}}$ $dt$

$e^{-|z|}$ = $\frac{1}{\pi}$ $\frac{e^{itz}}{1 + t^{2}}$  $dt$

On interchanging t and Z we have

$e^{-|t|}$ = $\frac{1}{\pi}$ $\frac{e^{itz}}{1 + t^{2}}$ $dz$

Therefore we get $\phi_{X}$(t) = $e^{-|t|}$
Given below are the important properties of cauchy distribution:

1) If $X_1 $,..., $X_n$ are separate and identically allocated random variables, each using a standard Cauchy distribution, then the sample mean $\frac{(X_1+ ... +X_n)}{n}$ has exactly the same standard Cauchy distribution.

2) The Cauchy distribution is definitely an infinitely divisible likelihood distribution. It can be a strictly secure distribution.

3) The standard Cauchy distribution coincides while using Student's t-distribution with one amount of freedom.

4) Cauchy distribution could be the only univariate distribution which is closed under linear fractional conversions with real coefficients.
If $X_{1}$ and $X_{2}$ are independent cauchy variates with parameters ($\lambda_{1}, \mu_{1}$) and ($\lambda_{2}, \mu_{2}$) respectively, then $X_{1} + X_{2}$ is a cauchy variate with parameters ($\lambda_{1} + \lambda_{2}, \mu_{1}+ \mu_{2}$)

Proof:
  
Consider $\phi_{X_{j}}(t) = exp(i\mu_{j}t - \lambda_{j}$|t|); (j = 1, 2)

Since X$_{1}, X_{2}$ are independent

$\phi_{X_{1}+ X_{2}}(t) = \phi_{X_{1}}(t) . \phi_{X_{2}}$(t)

= exp[it ($\mu_{1} + \mu_{2}) - (\lambda_{1}  + \lambda_{2}$)|t|]

By uniqueness theorem we can say that X$_{1}$ and X$_{2}$ are independent cauchy variates with parameters ($\lambda_{1}, \mu_{1}) and (\lambda_{2}, \mu_{2}$) respectively
Some of the problems based on cauchy distribution are explained below to have an easier understanding of the topic:

Example 1: Let X have a cauchy distribution. Find a p.d.f for X$^{2}$ and identify its distribution.

Solution:
Since X has a standard cauchy distribution, its p.d.f is

$f(x)$ = $\frac{1}{\pi}$.$\frac{1}{1 + x^{2}}$, -$\infty < x < \infty$

The distribution function G(.) of Y = $X^{2}$ is

G$_{Y}(y)$ = $P(Y \leq y)$ = $P(X^{2} \leq  y)$

= P(- $\sqrt{y} \leq X \leq \sqrt{y}$)

= $\int_{-\sqrt{y}}^{\sqrt{y}}$ $f(x)dx$

= 2 $\frac{1}{\pi}$ $\int_{0}^{\sqrt{y}}$ $\frac{dx}{1 + x^{2}}$

= $\frac{2}{\pi}$ $tan^{-1}(\sqrt{y})$, $0 < y < \infty$

The p.d.f. g$_{Y}$(y) is given by

$g_{Y}(y)$ = $\frac{d}{dy}$ [$G_{Y}(y)$] = $\frac{2}{\pi}$.$\frac{1}{(1 + y)}$.$\frac{1}{2\sqrt{y}}$

= $\frac{1}{\pi}$.$\frac{y^{-\frac{1}{2}}}{1 +y}$

= $\frac{1}{B(\frac{1}{2}, \frac{1}{2})}$.$\frac{y^{\frac{1}{2}-1}}{(1 + y)^{\frac{1}{2} + \frac{1}{2}}}$, y > 0

This is the pdf of beta distribution of second kind with parameters ($\frac{1}{2},\frac{1}{2}$)

Therefore X$^{2}$ $\sim$ $\beta_{2}$($\frac{1}{2}$,$\frac{1}{2}$)