Is probability mass function (pmf) unique?

Question

It is known that probability density function (pdf) is not unique, but can we say the same about pmf?

Also, what can be the possible example where pdf or pmf may not exist?

Answer 1

If you want a simple example where it may not exist, consider the discrete uniform on -inf to inf. That is each event on the Reals is equally likely.

Answer 2

If $X$ is a real random variable with cdf $F$, we call $X$ continuous is a function $f$ called a pdf exists for which $F(x)=\int_{-\infty}^x f(t) dt$ for all $x\in[-\infty,\,\infty]$. This $f$ is not unique, since changing its value at one point will leave each definite integral unchanged.

We call $X$ discrete if a function $f$ called a pmf exists for which $F(x)=\sum_{y\le x,\,f(y)\ne 0}f(y)$ for all $x\in[-\infty,\,\infty]$. This $f$ is unique, since changing $f(x_0)$ also changes $F(x_0)$. In this respect, addition is dissimilar to integration. The only way for integration to capture this feature of addition is to add a Dirac delta spike, but in that case we must renormalise the pdf. In fact, any pmf can be "translated" into something resembling a pdf, but the result won't be a true function.

The above cases are clearly very different. There are also choices of $F$ for which neither a pdf nor a pmf exists. One example is the Cantor distribution. A less complicated example is the distribution of the sum of two non-constant variables, one continuous and one discrete (e.g. the sum of a lifetime and a dice roll).