Definition:Median of Continuous Random Variable

Definition
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ have probability density function $f_X$.

A median of $X$ is defined as a real number $m_X$ such that:


 * $\ds \map \Pr {X < m_X} = \int_{-\infty}^{m_X} \map {f_X} x \rd x = \frac 1 2$

That is, $m_X$ is the first $2$-quantile of $X$.