survival-density-hazard relationship
Survival modeling has three faces of the same coin: the survival function S(x) (chance of living past x), the density f(x) (how the chance of death is spread across ages), and the force of mortality mu-x (the instantaneous danger at age x). They look different but describe identical information about a lifetime, and any one of them can be turned into the others. Knowing the relationships means you never have to start from scratch — you translate.
Three plain-language links tie them together. First, the density is how fast survival is being eaten away: f(x) = -S'(x), the (negative) slope of the survival curve. Second, the force is the density relative to the survivors still at risk: mu-x = f(x) / S(x) — deaths per unit time, divided by the pool still alive. Third, you recover survival by accumulating the force: S(x) = exp(- integral of mu from 0 to x). Put together, f(x) = S(x) times mu-x: the chance of dying right at age x equals the chance of reaching x, times the intensity of dying there.
These conversions are the daily grammar of survival analysis. A medical study might estimate the hazard directly from data and then need the survival curve to report a five-year survival rate; an actuary may start from a tabulated survival function and back out the force to compare two mortality laws. Because the three are interchangeable, you choose whichever is easiest to model or estimate, then convert to whatever your problem needs.
If S(x) = exp(-0.01x), then differentiating gives f(x) = 0.01 exp(-0.01x), and dividing gives mu-x = f(x)/S(x) = 0.01 — a constant force, the exponential lifetime.
From any one of S, f, mu you reach the other two: differentiate, divide, or integrate-then-exponentiate.
The relations only hold for a continuous lifetime model. In a life table sampled at whole ages you instead use discrete analogues (p-x, q-x) and a fractional-age assumption to fill the gaps in between.