Random variables and distributions

A random variable turns outcomes into numbers — the score on a die, the height of a person, the time until a component fails — so that arithmetic becomes possible. Its distribution records how the probability is spread across those numbers, and two summaries do most of the work: the expected value, which is the probability-weighted average, and the variance, which measures spread.

"Expected" is a badly chosen word. The expected value of a fair die is 3.5, and no roll ever produces 3.5. It is not the most likely outcome and need not be a possible outcome at all; it is the long-run average, the number the running mean approaches over many trials. Reading it as a prediction of any single trial is the standard misuse.

Continuous variables need one more adjustment. For something like height, measured exactly, the probability of any single value — precisely 180.000… cm — is zero, because there are uncountably many candidate values. Probability lives in intervals, and the density function gives probability per unit only after integration over a range. Among distributions the normal one has a special status, because the central limit theorem shows that sums of many independent small effects tend to it regardless of what the individual effects look like. That is why measurement error is usually normal. It is not, however, a universal law: incomes, city sizes and waiting times are famously not normal, and assuming normality because a sample is large is an error that better data never fixes.