age-at-death random variable

Imagine a newborn baby. We cannot know in advance how long it will live — it might die in infancy, or live to a hundred and three. Actuaries handle this uncertainty by treating the eventual age at death as a number we don't yet know but can reason about with probability. That unknown number is the age-at-death random variable, usually written X. It is the single most basic building block of every life-insurance and pension calculation.

Because X is random, we describe it not by one value but by the chances of each possibility: the probability that the newborn dies before age 1, before age 50, before age 90, and so on. X is a continuous, non-negative random variable. Its cumulative distribution function F(x) gives the probability of dying by age x, and its density f(x) describes how the chance of death is spread across the ages. For example, if F(70) = 0.4, then there is a 40 percent chance a newborn dies before reaching exact age 70.

Almost everything in survival modeling is a re-expression of X. The survival function is just 1 minus F(x); the force of mortality is built from f(x) and the survival function; the life table tabulates X for a whole imaginary cohort. Treating the lifetime as a random variable — rather than guessing a single 'life expectancy' number — is what lets actuaries price insurance fairly and hold the right reserves.