future-lifetime random variable
Insurance is almost never sold to newborns; it is sold to people who are already a certain age. The question that matters for a 65-year-old buying an annuity is not 'how long does a newborn live?' but 'how much longer will I, having already reached 65, live?' That remaining span of life is the future-lifetime random variable, written T(x): the time still left for a person known to be alive at exact age x.
Formally, T(x) = X - x given that X > x. The little phrase 'given that X > x' is everything: we have already survived to age x, so all the ways of dying before x are off the table, and the probabilities must be recomputed on the smaller world of lives that reached x. For a life aged 65, T(65) = 20 means surviving exactly 20 more years to age 85. The probability that T(x) exceeds t — that the person survives at least t more years — is the central quantity, written in actuarial notation as t-p-x.
T(x) is the workhorse of life contingencies. Present values of life annuities depend on how long T(x) lasts; the payout of a life insurance happens at the moment T(x) ends. Because it is conditional on survival to x, the same person's future-lifetime distribution changes every birthday — which is exactly why an annuity costs more for a 60-year-old than for an 80-year-old of the same expected payment.
For a life now aged 65, t-p-65 = Pr(T(65) > t) is the chance of living at least t more years; 10-p-65 = 0.85 says an 85 percent chance of reaching age 75.
Future lifetime is conditional: the probabilities are recomputed for someone known to have survived to age x.
T(x) is a conditional random variable, not the same as X minus a fixed number. The conditioning on survival to x is what makes survival models harder — and more interesting — than ordinary probability.