survival function
/ S(x) or s(x) /
The survival function answers the most natural question we can ask about a lifetime: what is the probability of still being alive at a given age? Written S(x), it gives the chance that a newborn survives past exact age x — that the age-at-death X is bigger than x. At birth S(0) = 1 (everyone is alive at the start), and it slides downward toward 0 as age climbs, because survival to ever-higher ages becomes ever less likely.
It is simply the mirror image of the distribution function: S(x) = 1 - F(x) = Pr(X > x). If 40 percent of newborns die before age 70, then S(70) = 0.6 — sixty percent are still alive at 70. A survival function is non-increasing (you cannot become more likely to be alive as you age) and runs from 1 down to 0. The probability of surviving from age x to age x+t is a ratio of survival values: t-p-x = S(x+t) / S(x), which neatly captures the 'given alive at x' conditioning.
The survival function is the everyday language of demography, medicine, and reliability engineering, not just life insurance — a 'five-year survival rate' for a cancer is exactly S(5) for that group of patients. In actuarial work it is the bridge between the abstract random variable X and the concrete life table, since the table's l-x column is just S(x) scaled up by a starting number of lives.
If S(60) = 0.80 and S(65) = 0.74, the probability that a newborn alive at 60 survives to 65 is 5-p-60 = S(65)/S(60) = 0.74/0.80 = 0.925.
Conditional survival is a ratio of survival-function values — the denominator strips out the lives lost before age x.
A common slip is to read S(x) as 'the fraction of a real population alive at x'. It is a probability for one life (or an expected fraction of an idealised cohort), not a guaranteed headcount — actual cohorts wobble around it because of random variation.