random variable

Suppose you roll two dice. The raw outcome is a pair like (3, 5), but usually you do not care about the pair itself; you care about the total, here 8. A random variable is exactly that step: it is a rule that reads off a number from whatever the experiment produced. It turns messy outcomes (heads-and-tails sequences, pairs of dice, points in a square) into ordinary numbers you can add, average, and compare.

Formally, a random variable X is a function from the sample space to the real line: it takes each outcome and returns a real number X(outcome). The dice example sends the outcome (3, 5) to the number 8. Once you have this function, the probability that X lands in some set of numbers is just the probability of all the outcomes that get sent there. For instance P(X = 8) is the probability of the event {(2,6), (3,5), (4,4), (5,3), (6,2)}, which is 5/36. So a random variable does not invent new randomness; it inherits its randomness from the underlying experiment and repackages it as numbers.

This is the central abstraction of probability: it lets us stop tracking complicated outcomes and instead study numerical quantities. Be careful with the name: a random variable is not a variable in the algebra sense (a placeholder for one unknown number) and it is not random in the sense of being lawless; it is a fixed function whose input is uncertain. We write random variables with capital letters (X, Y) and the particular values they might take with lowercase letters (x, y).