force of mortality (hazard rate)

The survival function tells you the chance of living past an age, but a doctor or actuary often wants something sharper: for a person alive right now, how intense is the danger of death at this very instant? That instantaneous rate of dying, per unit of time, for someone known to be alive at age x, is the force of mortality, written mu-x. Statisticians call the same idea the hazard rate, and it is used far beyond mortality — for machine failures, customer churn, anything that 'lasts until it ends'.

It is best read as an annualised rate, not a probability. If mu-65 = 0.012 per year, it means that, at the instant of turning 65, deaths are occurring at a pace of about 1.2 percent per year among the survivors. Over a very short slice of time dt, the chance of dying is roughly mu-x times dt. Crucially mu-x can exceed 1 even though it is 'per year', because it is a rate, not a probability — a force of 2 per year simply means a very steep, brief danger, not a 200 percent chance.

The force of mortality is the most informative single description of a life: it typically falls through childhood, bottoms out in the teens, then rises roughly exponentially through adult life (the famous Gompertz curve). Once you know mu at every age you can rebuild everything — survival function, density, life table — because survival from x to x+t is found by accumulating (integrating) the force over that span: t-p-x = exp(- integral of mu from x to x+t).