8.1 Joint probability mass functions

Recall that a random variable is simply a function from the sample space $\mathrm{\Omega }$ to the real numbers $R$, mapping each elementary outcome $\omega$ to a number. Formally, there is no reason not to define several such functions, $X_1,X_2,\mathrm{\dots }$ such that for each $\omega$ we get a set of numbers $X_1(\omega ),X_2(\omega ),\mathrm{\dots }$.

We present some basic theory for discrete random variables. Let $X$ and $Y$ be discrete random variables defined on the same sample space $\mathrm{\Omega }$. Their joint probability mass function is

As in earlier chapters, we concentrate on discrete random variables taking non-negative integer values.