91Ó°ÊÓ

According to the Definition 3.3.2, the \(\frac{1}{2}\) quantile of a random variable X is defined as \({F^{ - 1}}\left( {\frac{1}{2}} \right),\) where \(F\left( x \right)\) is the cumulative distribution function of X.

According to the Definition 4.5.1, every number m with the following property is called a median of the distribution of X:

\(\Pr \left( {X \le m} \right) \ge \frac{1}{2}\) and \(\Pr \left( {X \ge m} \right) \ge \frac{1}{2}\)

Let:

\(\begin{aligned}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{F^{ - 1}}\left( {\frac{1}{2}} \right) &= m\\\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow F\left( m \right) ^= \frac{1}{2}\\ \Rightarrow \Pr \left( {X \le m} \right) &= \frac{1}{2}\;\left( {{\rm{satisfies the first condition of median}}} \right)\end{aligned}\)

Again, let:

\(\begin{aligned}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{F^{ - 1}}\left( {\frac{1}{2}} \right) &= m\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow F\left( m \right) = \frac{1}{2}\\\,\,\,\,\,\,\, \Rightarrow \Pr \left( {X \le m} \right) &= \frac{1}{2}\;\\ \Rightarrow 1 - \Pr \left( {X > m} \right) = \frac{1}{2}\end{aligned}\)

\(\begin{aligned}{l} \Rightarrow \Pr \left( {X > m} \right) = \frac{1}{2}\\ \Rightarrow \Pr \left( {X \ge m} \right) \ge \frac{1}{2}\left( {{\rm{satisfies the second condition of median}}} \right)\end{aligned}\)

Thus, \(m = {F^{ - 1}}\left( {\frac{1}{2}} \right)\) satisfies both the conditions of being the median. Hence, m is the median.