Can a set of data have multiple medians?

Question

I'm looking at a probability distribution where the cumulative probability distribution for a random variable $X_2$ is exactly 0.5. Does this mean that the distribution has multiple medians?

Answer 1

It depends on what you define as "median". What would you consider the median of $\{0,2\}$ to be? For a real-valued random variable, if the cumulative density function is constant at $0.5$ over an open interval, then it means that the possible values jump across a gap. It is reasonable to define the median as the midpoint of the gap, in which case there will always only be one median if any. For discrete distributions, you may want to consider the median of $\{0,2\}$ to be both $0$ and $2$, but this would not extend naturally to non-discrete distributions, because what would the median of $X$ be where $X$ takes a value in $[0,1)$ with probability $\frac{1}{2}$ and takes a value in $(2,3]$ with probability $\frac{1}{2}$?