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