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.