Discrete vs Continuous vs Random Variables Discrete vs Continuous vs Random Variables A random variable , formally, is a function $X: \Omega \to \mathbb{R}$ for some set $\Omega$. You can think of the set $\Omega$ as consisting of possible outcomes of a random experiment, and for any given input, $X$ tells you some measurement about the outcome. For example, if our random experiment is flipping a coin six times, $\Omega$ is the set $$\{(H,H,H,H,H,H), (H,H,H,H,H,T), (H,H,H,H,T,H), \dots, (T,T,T,T,T,T)\}.$$ A random variable $X$ might tell us the number of heads in the six coin flips, or it might tell us the number of runs of tails in the six coin clips. (If you want to be very precise, a random variable is a measurable function from measure space $(\Omega, \mathcal{F}, P)$ to some other set, but typically the range is $\mathbb{R}$, or perhaps $\mat