The Bathtub Curve and Failure-Rate Modelling

The hazard rate of a component at time $t$ is the instantaneous chance it fails right now, given it has survived up to $t$. Plotted over the lifetime of a typical manufactured product — a hard drive, a light bulb, a mechanical part — this rate traces a distinctive bathtub curve: high at the very start, dropping quickly, staying low and roughly flat for a long middle stretch, then rising again near the end of the item's useful life.

The three phases have different causes and are usually modeled separately rather than with one formula. Early failures ("infant mortality") come from manufacturing defects and design flaws that show up almost immediately — this is why products get burned in or tested before shipping, to push customers past this steep part of the curve before they ever see it. The flat middle is dominated by random, memoryless failures unrelated to age, often modeled with a constant hazard rate (an exponential lifetime distribution), which is why "useful life" failure rates are often quoted as a single number. The rising tail reflects genuine wear-out — fatigue, corrosion, accumulated stress — typically modeled with a Weibull distribution whose shape parameter is set above one to produce an increasing hazard.

The same shape shows up informally outside hardware: a new trading strategy is at highest risk of failure right after launch (bugs, misconfigured risk limits), settles into a stable failure rate during its productive life, and risk often rises again as market structure shifts underneath an aging, unmaintained model.