Q:

Median of random variable times constant

Let $\xi$ be a random variable with the median $m_ξ$. Is it true that  $m_{\varepsilon\xi} = \varepsilon m_\xi, \forall \varepsilon \in \mathbb{R}$?

A:

In general $m_{\xi}$ serves as median of $\xi$ iff: $$P\left(\xi\leq m_{\xi}\right)\geq0.5\wedge P\left(\xi\geq m_{\xi}\right)\geq0.5\tag1$$
Sidenote: a median is not necessarily unique. See also this question and its answer.
So the question can rephrased as: does $(1)$
imply that: $$P\left(\epsilon\xi\leq\epsilon m_{\xi}\right)\geq0.5\wedge P\left(\epsilon\xi\geq\epsilon m_{\xi}\right)\geq0.5\text{ for every }\epsilon\in\mathbb{R}$$?
The answer on this is: yes (also if $\epsilon\leq0$).

