complete and curtate life expectancy

Life expectancy is the average number of years a person can expect still to live, given their current age — the single most quoted summary of a life table. We say 'a 65-year-old has a life expectancy of 20 years' to mean the average remaining lifetime of people now aged 65. It is the mean (expected value) of the future-lifetime random variable, so it summarises a whole distribution of possible lifespans in one number.

There are two versions matching the two ways of measuring future lifetime. The complete life expectancy, written e-circle-x, is the average of the exact remaining time T(x) — found by integrating the survival probabilities, or just summing them across all future ages and adding a half-year correction. The curtate life expectancy, written e-x, is the average number of whole years completed; it is the simple sum p-x + 2-p-x + 3-p-x + ... of the survival probabilities. The two differ by roughly half a year: e-circle-x is about e-x + 0.5, the missing fraction of the final year of life.

Practically, life expectancy underpins pension design, retirement planning, and public-health comparisons, and it is what people mean by 'living longer'. But it is an average, not a promise: half the cohort outlives it and half does not, and the spread around it is wide. A common misreading is to treat 'life expectancy at birth of 80' as a personal guarantee — it is heavily pulled down by infant deaths, and a person who has already reached 65 typically has a far higher expected age at death than 80.