2.3 Random variable

A random ``variable'' is actually a function that has a domain of experiment outcomes and a range of numbers (usually counting numbers or natural numbers). A random variable is often denoted as $X$ in discussions.

Let us consider the experiment of attempting to throw a rock into two buckets. Let us further color the buckets red and blue. The experiment has three outcomes: the rock lands in the red bucket, the rock lands in the blue bucket, or it lands outside of both buckets. The set of possible outcomes is usually denoted by uppercase omega ($\Omega$ ). In this case, $\Omega = \{{\rm red bucket}, {\rm blue bucket}, {\rm outside buckets}\}$ .

The random variable function maps $\Omega$ to numeric values. In our case, we can arbitrarily define our random variable as $X = \{({\rm red bucket},0), ({\rm blue bucket},1), ({\rm outside buckets},2)\}$ .

Sometimes, we don't really need a fancy random variable. For example, in the case of dice rolling, the random variable (function) is merely $X = \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}$ .

In general, a random variable function is just a way to enumerate all possible experiment outcomes as numeric values. It is particularly useful for continuous probability distributions.