Solved – Median of the half-Cauchy distribution

Question

The probability density function $f(x)$ of a Cauchy distributed random variable $x$ is given by:

$$f(x; x_0,\gamma) = { 1 \over \pi \gamma } \left[ { \gamma^2 \over (x – x_0)^2 + \gamma^2 } \right]$$

The mean and variance of $x$ are undefined however its median is defined as $x_0$. A half-Cauchy distributed random variable $y$ is given by the absolute value of a Cauchy distributed random variable with $x_0 = 0$. Its probability density function is given by:

$$f(y;\gamma) = { 2 \over \pi \gamma } \left[ { \gamma^2 \over y^2 + \gamma^2 } \right] \quad \text{for }y \geq 0$$

The mean and variance of $y$ are undefined however my question is whether the median of $y$ is defined and if so what is the median given by?

Answer 1

From any symmetric distribution, you can define a half-distribution simply by doubling the part of the distribution above the symmetry point (and eliminating the part below the symmetry point). Then it is clear that the upper quartile of original distribution will be the median of the half-distribution.

For the standard Cauchy this gives (using R, you can do it symbolically using the formula for the cdf):

 qcauchy(0.75)
[1] 1

Or the Cauchy with symmetry point 10 and scale parameter 2:

 qcauchy(0.75, 10, 2)
[1] 12