91Ó°ÊÓ

According to Example 4.1.8, a random variable X is said to follow the Cauchy distribution if it has the pdf \(f\left( x \right) = \frac{1}{{\pi \left( {1 + {x^2}} \right)}};\;{\rm{for}}\; - \infty < x < \infty .\)

First, find the cumulative distribution function of X as follows:

\(\begin{aligned}{}F\left( x \right) &= \int\limits_{ - \infty }^x {\frac{1}{{\pi \left( {1 + {x^2}} \right)}}dx} \\ &= \frac{1}{\pi }\int\limits_{ - \infty }^x {\frac{1}{{1 + {x^2}}}dx} \\& = \frac{1}{\pi }\left( {{{\tan }^{ - 1}}x} \right)_{ - \infty }^x\\ &= \frac{1}{\pi }{\tan ^{ - 1}}x + \frac{1}{\pi } \times \frac{\pi }{2}\\ &= \frac{1}{\pi }{\tan ^{ - 1}}x + \frac{1}{2}.\end{aligned}\)

Now, the median of X is defined as:

\(\begin{aligned}{}F\left( m \right) &= \frac{1}{2}\\\frac{1}{\pi }{\tan ^{ - 1}}m + \frac{1}{2} &= \frac{1}{2}\\{\tan ^{ - 1}}m &= 0\\m &= 0.\end{aligned}\)

Hence, for a standard Cauchy distribution, the median is 0