Multiple, One (?), or No Medians for discrete

Question

I am a bit confused on how medians are calculated for discrete distributions. I understand that there can be multiple medians or none at all. Can there be one median for a discrete distribution? Can you provide examples of all three (if possible) and quick notes on how I would solve for each respective 'm'? Thank you.

Answer 1

https://en.wikipedia.org/wiki/Median

There should be at least one median for any distribution.

In fact it should be common for a discrete distribution to have a unique median. Note that the CDF of a discrete distribution $F$ is a step function consists of jumps which corresponding to the point masses.

The way we find the median is that we look at the horizontal line $y = 0.5$ on the graph $y = F(x)$. When there is a jump over this line (i.e. no intersection), then the median corresponding to the location of this point mass, which means that the median is unique. There are multiple medians only when $F(x) = 0.5$ has a solution, i.e. the function overlap with the line and all intersecting points will satisfy the definition of median.