Definition:Median of Continuous Random Variable

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This page is about Median of Continuous Random Variable. For other uses, see Median.


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 $X$.


Care should be directed towards the uniqueness of medians.

For example, if $\map {f_X} x = 0$ on some closed real interval $\closedint a b$ of $x$, we may have $\ds \int_{-\infty}^{m_X} \map {f_X} x \rd x = \frac 1 2$ for all $m_X \in \closedint a b$.