> 
> (Textbook accepted) Definition: The median of a random variable X is any real value $m$
>  s.t. $P(X \geq m)\geq\frac{1}{2}$ and $P(X \leq m)\geq \frac{1}{2}$.
> 
> 
> 

I've come up with 4 alternative definitions of the "median." I'm writing to ask whether those would also work against all counterexamples. 

$(1)$

A "median" of a random variable X is any real value $m$
 s.t. $P(X \geq m) < \frac{1}{2}$ and $P(X \leq m) < \frac{1}{2}$.

$(2)$

A "median" of a random variable X is any real value $m$
 s.t. $P(X \geq m) < \frac{1}{2}$ and $P(X \leq m) \leq \frac{1}{2}$
.

$(3)$
A "median" of a random variable X is any real value $m$
 s.t. $P(X \geq m) \leq \frac{1}{2}$ and $P(X \leq m) \leq \frac{1}{2}$

$(4)$

A "median" of a random variable X is any real value $m$
 s.t. $P(X \geq m) \leq \frac{1}{2}$ and $P(X \leq m) < \frac{1}{2}$