Reliability function and hazard rate

The formula

R(t) = P(T > t) = 1 − F(t)

Reads: the probability a unit's life T exceeds t — the complement of the failure CDF ^[2]. Starts at 1, only goes down.

f(t) = −dR/dt

Reads: the failure density is how fast the survival curve is dropping — failures per unit time as a fraction of the original population.

h(t) = f(t) / R(t) = −d/dt ln R(t)

Reads: the hazard rate is the failure density divided by the fraction still alive — failures per unit time among the survivors. This is the "instantaneous failure rate" on a data sheet.

R(t) = exp( −∫₀ᵗ h(u) du ) = e^(−H(t))

Reads: any hazard curve, integrated, gives you back the survival curve. H(t) is the cumulative hazard. This pair works for every life distribution, not just the exponential.

Two special cases carry most of the load in engineering:

Exponential:  h(t) = λ (constant),   R(t) = e^(−λ·t),   MTBF = 1/λ
Weibull:      h(t) = (β/η)·(t/η)^(β−1),   R(t) = exp( −(t/η)^β )

Weibull's shape parameter β sets the hazard's direction: β < 1 falling (infant mortality), β = 1 constant (it collapses to the exponential), β > 1 rising (wear-out) ^[9].

Where you meet it

Parts-count reliability prediction. A predictions engineer runs the BOM through MIL-HDBK-217-style models and hands you a λ for the box in FITs — failures per 10⁹ device-hours ^[1]^[13]. Everything downstream of that number — MTBF, sparing, mission reliability — is this vocabulary.
Life test data on the bench. Run twenty units to failure, plot the failure times on Weibull paper, and the fitted β tells you why they're failing: β well under 1 says workmanship escapes, β around 3 says a wear-out mechanism. Same math behind the L10 life on a bearing catalog page.
The review board argument about MTBF. Somebody presents a 2,000,000-hour MTBF for a 5-year mission and somebody else asks what the reliability actually is at end of life. R(t) = e^(−λt) settles it in one line — and exposes whether the constant-rate assumption was ever justified.
Burn-in and screening decisions. Whether a 48-hour burn-in buys you anything is entirely a question about whether h(t) is decreasing at t = 0. If the hazard is flat, burn-in just consumes life.

How it works

The hazard rate is a conditional rate, and that conditioning is the whole point. f(t) answers "what fraction of the units we started with fail near time t." h(t) answers "given this unit made it to t, what's its failure rate right now." For a fleet of aging actuators, f(t) eventually falls simply because few units are left; h(t) keeps climbing. The one you feel as the owner of a surviving unit is h(t).

A hazard rate is not a probability. Its units are 1/time, and nothing stops h(t)·Δt from being read wrong. h(t)·Δt approximates the probability of failure in the next Δt given survival so far, and only for small Δt. A hazard of 2 per year does not mean 200% chance of failure this year — it means R = e^(−2) ≈ 0.135 if the rate holds.

The bathtub curve is a statement about h(t), not f(t): early-life failures from defects (falling hazard), a long flat useful-life region, then wear-out (rising hazard) ^[1]. The exponential model — the one behind every bare MTBF number — assumes you live entirely in the flat middle. That assumption is the standing criticism of handbook predictions: constant-rate models fit curves without touching the physics of failure, and they can't see either infant mortality or wear-out ^[12].

The MTBF trap, with numbers. λ = 500 FIT means λ = 5·10⁻⁷ per hour, so MTBF = 2·10⁶ hours — about 228 years. That does not mean the unit lasts 228 years; it means that over 10,000 operating hours, R = e^(−0.005) ≈ 0.995. Run any exponential unit for exactly one MTBF and its survival probability is e^(−1) ≈ 0.368. Nearly two-thirds of units fail before the MTBF. If a briefing chart implies MTBF is a lifetime, the chart is wrong.

Worked Weibull check: β = 2, η = 1,000 h gives R(500) = exp(−(0.5)²) = e^(−0.25) ≈ 0.779 and h(500) = (2/1000)·(0.5) = 0.001 per hour — a hazard that has doubled since t = 250 and keeps rising linearly. A single "failure rate" number for this unit is meaningless without saying when.

One more mistake worth naming: hazard rate applies to non-repairable items (or to a component's first failure). For a repairable system logging failures over calendar time, the failure intensity (rate of occurrence of failures) is a different animal — fixing a box resets nothing about the ones still in the field, and pooling repair events into a Weibull fit of "times to failure" produces confident nonsense. Check which question you're asking before you pick the math.

Hazards add for a series system of independent parts: h_sys(t) = Σ hᵢ(t). That additivity is the entire justification for parts-count prediction — and it holds only in series logic with independence. Redundancy breaks it; then you go back to R(t) and do the block algebra.

History

The survival curve is older than engineering reliability by three centuries, and it started with death records. In 1662 London haberdasher John Graunt published his Observations on the city's Bills of Mortality and constructed what is credited as the first life table — survival probabilities by age, reasoned out of records that didn't even list age at death ^[3]^[4]. In 1693 Edmond Halley, working from Caspar Neumann's meticulous records for the city of Breslau, published a proper life table relating mortality to age, which let annuities be priced by the buyer's age and set actuarial science in motion ^[4]^[5]. In 1825 the actuary Benjamin Gompertz found the pattern hiding in those tables: the force of mortality — what engineers now call the hazard rate — grows roughly exponentially with age ^[6]^[7]. Humans are a wear-out population; Gompertz measured the β.

The engineering side arrived through broken parts. The distribution now carrying Waloddi Weibull's name was identified by Maurice Fréchet in 1927 and used by Rosin and Rammler in 1933 for particle sizes; Weibull worked it out in detail in 1939 studying the strength of materials, then made its case to engineers in his 1951 paper "A statistical distribution function of wide applicability," walking through seven case studies ^[8]^[9]. Postwar military electronics turned failure rates into an industry: RCA's TR-1100 report from the 1950s grew into MIL-HDBK-217, the reliability-prediction handbook whose part-failure-rate models — and whose constant-hazard assumption — reliability engineers have used and argued about ever since; its last full update was Revision F, Notice 2, in 1995 ^[12]^[13]. The capstone came from statistics: David Cox's 1972 paper "Regression Models and Life-Tables" showed how to model a hazard as a baseline curve scaled by covariates without ever specifying the baseline — the proportional hazards model that now dominates survival analysis in medicine and shows up in accelerated life testing ^[10]^[11].

Related tools

/tools/bearing-life-l10 — L10 is the 10th-percentile life, a point read straight off a Weibull reliability curve
/tools/half-life-decay — same exponential-survival math with λ relabeled as a decay constant
/tools/arrhenius-rate — the acceleration model behind temperature-accelerated life tests that feed hazard estimates
/tools/battery-life — runtime estimation that pairs with failure-rate math in mission-duration trades

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →