3. Probability distributions for reliability engineering

Reliability analysis begins with a model of how time-to-failure behaves. Four distributions dominate engineering practice: the exponential, the Weibull, the lognormal, and the gamma. Each encodes different assumptions about failure mechanisms, and each leads to different estimates of reliability, hazard rate, and maintenance intervals. Choosing the wrong family is not a cosmetic error — it can shift a predicted B10 life by an order of magnitude. This section develops the four models, their parameters, their hazard behavior, and how to fit them to field data.

Exponential: constant hazard

The exponential distribution assumes a constant failure rate λ. Its density and reliability functions are:

f(t) = λ · exp(−λt)
R(t) = exp(−λt)
h(t) = λ            (constant)
MTTF = 1 / λ

The defining property is memorylessness: a component that has survived 10,000 hours has exactly the same remaining failure probability as a new one. This is realistic for random, externally driven failures — electronic components in their useful-life phase, software faults triggered by rare input combinations, or events caused by random overstress. It is not realistic for wear-out or early-life burn-in.

Field estimation is simple. If r failures are observed over a total accumulated time T (including units still running), the maximum-likelihood estimate is λ̂ = r / T. For example, 6 failures across 12 pumps accumulating 48,000 operating hours gives λ̂ = 6 / 48,000 = 1.25 × 10⁻⁴ per hour, or MTTF ≈ 8,000 h. The 95% confidence interval follows a chi-square relation: [χ²(0.025, 2r) / 2T, χ²(0.975, 2r+2) / 2T].

Weibull: the workhorse

The two-parameter Weibull generalizes the exponential by adding a shape parameter β that captures the failure mode:

R(t) = exp(−(t/η)^β)
h(t) = (β/η) · (t/η)^(β−1)
MTTF = η · Γ(1 + 1/β)

Here η is the scale parameter (characteristic life, where R = 0.368) and β is the shape parameter. The interpretation of β is the single most useful diagnostic in reliability engineering:

• β < 1 — decreasing hazard: infant mortality, manufacturing defects, contamination.
• β = 1 — constant hazard: the Weibull reduces exactly to the exponential.
• β > 1 — increasing hazard: wear-out, fatigue, corrosion, cumulative damage.
• β ≈ 3.5 — approximates a normal distribution, typical of bearing fatigue.

Fitting is done on the Weibull probability plot: transform to ln(−ln R(t)) versus ln t, and a straight line yields β as slope and η from the intercept. With censored field data (units removed before failure), use median-rank or maximum-likelihood estimation rather than simple regression. A practical example: 20 field returns of a gearbox show failures at 1,200, 1,850, 2,400, 3,100, 3,900, 4,600, 5,500, 6,800, 8,200, and 9,700 hours, with 10 units still running at 12,000 h. A Weibull fit gives β ≈ 1.9 and η ≈ 8,600 h, indicating clear wear-out. The B10 life (time at which 10% fail) is η·(−ln 0.9)^(1/β) ≈ 8,600 × 0.376 ≈ 3,230 h — a defensible replacement interval.

Lognormal: multiplicative degradation

If the logarithm of time-to-failure is normally distributed, the lifetime is lognormal. It arises naturally when failure results from the product of many small multiplicative effects — crack growth, corrosion pit deepening, or semiconductor degradation. Its hazard rate rises to a peak and then decreases, which distinguishes it from the Weibull.

f(t) = 1/(t·σ·√(2π)) · exp(−(ln t − μ)² / (2σ²))
MTTF = exp(μ + σ²/2)
Median = exp(μ)

Here μ and σ are the mean and standard deviation of ln t, not of t itself. This is a common source of error: reporting "mean life" as exp(μ) understates MTTF whenever σ is large. Lognormal is the preferred model for repair times and for fatigue-crack-initiation data. Fit it by taking natural logs of the failure times and applying standard normal methods; with censoring, use MLE.

Gamma: cumulative damage and repair

The gamma distribution models the time to the k-th event in a Poisson process, making it natural for systems that fail only after accumulating k shocks or damage increments. It also fits repair-time and standby-redundancy problems.

f(t) = (λ^k · t^(k−1) · exp(−λt)) / Γ(k)
MTTF = k / λ
h(t) → λ as t → ∞   (increasing if k > 1)

With integer k, the gamma is the Erlang distribution. Its hazard is increasing for k > 1 and decreasing for k < 1, giving it similar flexibility to the Weibull but with a different tail shape. The gamma is especially useful in Bayesian reliability because it is the conjugate prior for the exponential rate λ — a theme developed later in this course.

Choosing and validating the model

Model selection should be driven by failure physics first, then confirmed statistically. A practical workflow:

1. Plot the data. Build a Weibull probability plot. If points fall on a straight line, Weibull is adequate. Curvature suggests a mixture (two failure modes) or a different family.
2. Check the hazard trend. Use a cumulative-hazard plot (Nelson-Aalen) or a trend test. A concave cumulative hazard implies decreasing h(t); convex implies increasing.
3. Compare fits. Compute AIC or BIC across candidate models. A difference of less than 2 is inconclusive; prefer the simpler model.
4. Validate with holdout data. Predict failures in a later time window and compare with observed counts.

Field data example: comparing models on the same dataset

Consider 15 identical solenoid valves with recorded failure times (hours): 2,100; 2,600; 3,050; 3,400; 3,900; 4,250; 4,800; 5,300; 5,900; 6,400; 7,100; 7,800; 8,600; 9,500; 10,400. No censoring. Fitting each model by MLE gives the following approximate results (illustrative, computed from the stated data):

The Weibull achieves the lowest AIC, and its β = 1.62 confirms wear-out. The exponential is rejected because its AIC is 6.7 higher and the hazard is clearly not constant. The lognormal and gamma are competitive but offer no physical advantage here. The engineering conclusion: schedule preventive replacement near the B10 life of approximately 2,900 h, and monitor for a shift in β that would signal a new failure mode.

These four distributions form the vocabulary of parametric reliability. The exponential handles random events, the Weibull handles monotone hazard trends, the lognormal handles multiplicative degradation, and the gamma handles cumulative-damage and Bayesian conjugate settings. The next step is to fit them rigorously to censored field data and quantify the uncertainty in their parameters.