## Cauchy Distribution

A continuous random variable $X$ is said to follow Cauchy
distribution with parameters $\mu$ and $\lambda$ if its probability density function is given by
```
$$
\begin{equation*}
f(x; \mu, \lambda) =\left\{
\begin{array}{ll}
\frac{\lambda}{\pi}\cdot \frac{1}{\lambda^2+(x-\mu)^2}, & \hbox{$-\infty < x< \infty$;} \\
& \hbox{$-\infty < \mu< \infty$, $\lambda>0$;} \\
0, & \hbox{Otherwise.}
\end{array}
\right.
\end{equation*}
$$
```

## Mean and variance of Cauchy Distribution

Cauchy distribution does not possesses finite moments of order greater than or equal to 1. Hence, mean and variance does not exists for Cauchy distribution.

## Median of Cauchy Distribution

The median of Cauchy distribution is $\mu$.

## Mode of Cauchy Distribution

The mode of Cauchy distribution is $\mu$.

## Quartiles of Cauchy Distribution

The Quartiles of Cauchy distributions are $Q_1=\mu-\lambda$, $Q_2=\mu$ and $Q_3=\mu+\lambda$.

## Quartile Deviation of Cauchy distribution

```
$$
\begin{equation*}
QD = \frac{Q_3-Q_1}{2}
=\frac{\mu+\lambda-(\mu-\lambda)}{2}=\lambda.
\end{equation*}
$$
```

## Distribution Function of Cauchy Distribution

The distribution function of Cauchy distribution is
```
$$
\begin{equation*}
F(x) =\frac{1}{\pi}\tan^{-1}\bigg(\frac{x-\mu}{\lambda}\bigg) + \frac{1}{2}.
\end{equation*}
$$
```

## MGF of Cauchy Distribution

Moment generating function of Cauchy distribution does not exists

## Characteristics Function of Cauchy Distribution

```
$$
\begin{aligned}
\phi_{X}(t)& =e^{i\mu t -\lambda|t|}.
\end{aligned}
$$
```