Random variables:

We now need some formal apparatus for talking about events. We introduce the notion of a random variable as a formal reflex of the idea of a trial. Call it X (we systematically use upper-case for random variables).

Random variables represent what you know about a trial before you have seen its outcome. Thus having a random variable corresponding to a coin-toss is the same as tossing the coin, catching it and putting it on the table, but not yet looking which of the two possible sides is face up.

In the case of a random variable X which ranges over a sample space of k mutually exclusive outcomes, we notate the outcomes as x_i where $1 \leq i \leq k$ (we systematically use subscripted lower-case versions of letters to indicate possible outcomes of the trial corresponding to the random variable indicated by the upper-case version of the same letter).