Hazard Rate and Residual Lifetime

A trading server has been running for 400 days without failure. Does that make it more likely to fail tomorrow (it's "wearing out"), less likely ("proven reliable"), or exactly as likely as a brand-new server? The answer depends entirely on the shape of the component's hazard rate — how the instantaneous risk of failure changes with age. Getting this wrong leads to replacing components too early (wasting money) or too late (risking outages), and it's the same question underlying credit risk (does a seasoned loan default less than a new one?) and options (does an at-the-money option's chance of a big move change as expiry approaches?).

An analogy: an aging car versus a memoryless lottery

An old car with 150,000 miles is intuitively more likely to break down next month than one with 5,000 miles — mechanical wear accumulates, so the hazard rate increases with age. Compare a lottery ticket that draws a random number daily and "fails" (wins) if it matches yours: its chance of winning tomorrow doesn't depend on how many days it's gone without winning, because each day is a fresh, independent draw — a constant, age-independent hazard rate. Real failure processes span this range, from increasing hazard (wear) to constant hazard (random shocks) to even decreasing hazard (early "infant mortality" defects that, once survived, mark a genuinely good unit).

The formulas, one piece at a time

Let $T$ be the random lifetime of a component, with survival function $S(t) = P(T > t)$ (probability of lasting past age $t$) and density $f(t)$. The hazard rate at age $t$ is

$$h(t) = \frac{f(t)}{S(t)},$$

which is the instantaneous rate of failure at age $t$, conditional on having survived to age $t$. In plain English: $h(t)$ answers "given it's lasted this long, what's the failure rate right now?" — not the overall failure rate from birth, but the conditional rate given survival so far. The residual lifetime given survival to age $t$ is $T - t \mid T > t$, and its expected value, $\mathbb{E}[T - t \mid T > t]$, is the expected remaining life. For the exponential distribution specifically, $h(t)$ is constant — the memoryless property — meaning $\mathbb{E}[T - t \mid T > t] = \mathbb{E}[T]$ for every $t$: a component that's already survived to age $t$ has exactly the same expected remaining lifetime as a brand-new one, however large $t$ is.

Worked example 1: the memoryless exponential server

Suppose server failures follow an exponential distribution with rate $\lambda = 1/500$ per day (mean lifetime 500 days), modeling failures driven by random external shocks (power surges, cosmic-ray bit flips) rather than wear. The hazard rate is constant: $h(t) = \lambda = 1/500 = 0.002$ per day, at every age $t$. A server that's survived 400 days has exactly the same expected remaining lifetime, $\mathbb{E}[T] = 500$ days, as a server installed today. This is genuinely counterintuitive — "it's been running fine for over a year" provides zero information about how much longer it will last, under this model.

Worked example 2: increasing hazard, a Weibull-style component

Now suppose a different component's lifetime follows a Weibull distribution with shape parameter $k = 2$ (hazard rate increasing linearly in age) and scale parameter matched so $h(t) = 2t/500^2$ per day. At $t=100$ days, $h(100) = 200/250{,}000 = 0.0008$ per day. At $t=400$ days, $h(400) = 800/250{,}000 = 0.0032$ per day — four times higher, reflecting genuine wear accumulation. Here, unlike the exponential case, a component that's survived 400 days genuinely faces a higher near-term failure risk than a new one, and the expected residual lifetime shrinks with age rather than staying constant — exactly the mechanical-wear intuition, formalized.

Drag the rate parameter above and watch how the curve's decay shape relates to a constant hazard rate — a steeper exponential decay corresponds to a higher constant $\lambda$, and the memoryless property holds regardless of the rate chosen.

What this means in practice

Hazard rate shape drives real decisions: whether "seasoned" loans (survived several years without default) are safer or riskier going forward than freshly originated ones, whether a market-making algorithm's fill probability should be modeled as memoryless or age-dependent, and whether preventive maintenance schedules make sense at all — replacing a memoryless component early buys nothing, since it's exactly as likely to fail tomorrow as one installed today, while replacing a wearing component early can meaningfully cut failure risk.