What do we know about random variables? They are functionals that map events (which are subsets of a sigma-algebra of some sample space) to real numbers. The name random variable is a little bit misleading because they are neither random nor variable. Their name originates from the fact that mathematicians use them to map the possible outcomes of random and/or abstract experiments to the real numbers for the purpose of being able to do arithmetic. Random variables must also be measurable, and in particular measurable with respect to the associated sigma-algebra. But random variables also generate something rather peculiar.

#### Probability Mass Function

When one observes a random variable in action one sees that an experiment is performed and the random variable carries that experiment across to the domain of the real numbers, providing you with a bone fide real number to go along with that experiment. Do the experiment again and you get a new, perhaps different, real number. Do it again, another real number, and so on. Suppose, as you are doing this, you record the number you got and tallied its frequency.