What is a Random Variable?

A function that maps outcomes to numbers we can do math with.

A random variable $X$ is a function from the sample space $\Omega$ to the real numbers. It assigns a numerical value to each outcome of an experiment, turning raw outcomes into something we can sum, average, and analyze.

A concrete example: flip three coins. The sample space has $8$ equally likely strings (HHH, HHT, HTH, …). Define $X$ to be the number of heads. Then $X$ takes values in $\{0, 1, 2, 3\}$ with probabilities

P(X = 0) = \tfrac{1}{8}, \quad P(X = 1) = \tfrac{3}{8}, \quad P(X = 2) = \tfrac{3}{8}, \quad P(X = 3) = \tfrac{1}{8}

Probability theory is much easier to do on numbers than on raw outcomes. Random variables let us compute expectations, variances, correlations, and apply the full machinery of calculus and linear algebra to uncertain quantities. In quant finance, the daily return of a stock, the number of trades in a session, and the loss of a portfolio are all random variables.