the survival function

Where the cdf asks how much probability lies at or below a value, the survival function asks the opposite: how much lies strictly above it? If X is the lifetime of a lightbulb, the survival function S(t) is the probability the bulb is still working past time t, that it has survived to t. It is the natural language whenever you care about lasting, exceeding, or holding out, rather than falling short.

Formally it is just the complement of the cdf: S(x) = P(X > x) = 1 - F(x). So the two carry the same information, mirror images of each other. The survival function starts at 1 on the far left (everything survives the start) and decreases to 0 on the far right (eventually nothing survives), exactly the reverse of the cdf climbing from 0 to 1. To get the probability a value falls in (a, b], subtract: P(a < X <= b) = S(a) - S(b).

Survival functions are the everyday tool in reliability engineering, medicine, and insurance, where the question is almost always how long something lasts: the chance a patient lives beyond five years, a machine runs past its warranty, or a policyholder outlives a payout schedule. Working with S directly is often cleaner than with F because the interesting tail probabilities (rare long-lasting cases) are right there. The survival function is also the gateway to the hazard rate, which describes the instantaneous risk of failing given survival so far.