4. Probability distributions for time-to-failure

When an engineering test ends with some units failed and others still running, the raw failure times alone do not describe the underlying life behavior. A probability distribution converts those observations into a model that can answer questions such as: What fraction will survive to 1,000 hours? What is the B10 life? How does the hazard rate change with age? This chapter covers the three distributions most frequently used for time-to-failure data — exponential, Weibull, and lognormal — and shows how to estimate their parameters when the data are censored.

Why Distribution Choice Matters

Each distribution encodes a different assumption about how failure risk evolves over time. The exponential distribution assumes a constant hazard rate: the probability of failure in the next hour is the same regardless of how long the unit has already operated. This is appropriate for random, memoryless failures such as certain electronic component failures, but it is rarely adequate for wear-out or fatigue mechanisms.

The Weibull distribution generalizes the exponential by adding a shape parameter β. When β < 1, the hazard rate decreases with time (infant mortality). When β = 1, it reduces to the exponential. When β > 1, the hazard rate increases (wear-out). This flexibility makes Weibull the default choice for many mechanical and electromechanical reliability studies.

The lognormal distribution arises naturally when failure is driven by a multiplicative degradation process — crack growth, corrosion, or cumulative damage where each increment is proportional to the current state. Its hazard rate rises to a peak and then decreases, which suits some fatigue and semiconductor failure mechanisms.

Parameter Estimation from Censored Data

Test data are almost never complete. Units may be removed before failure (right-censored), or testing may stop at a fixed time (time-censored). Ignoring censoring and using only the failed units' times biases the estimate toward shorter lives. The correct approach uses the likelihood function that accounts for both failed and censored observations.

For right-censored data with failure times t_i and censoring times c_j, the likelihood for a Weibull distribution is:

L(β, η) = ∏_{failed i} (β/η)(t_i/η)^{β-1} exp[-(t_i/η)^β]
          × ∏_{censored j} exp[-(c_j/η)^β]

Maximizing this likelihood — typically by taking logarithms and solving numerically — yields the maximum likelihood estimates (MLEs) for β and η. For the exponential, the MLE of λ is simply the total number of failures divided by the total time on test (including censored units' operating time). For the lognormal, the MLEs of μ and σ are computed from the logarithms of the failure times, with censored observations contributing to the survival term.

A practical alternative to MLE is probability plotting. For the Weibull, plotting ln[−ln(1−F(t))] against ln(t) should yield a straight line if the Weibull model fits; the slope estimates β and the intercept gives η. Median rank positions are used for the plotting positions of the failed units, and censored units are plotted as suspended items. This graphical method is valuable for assessing goodness of fit and spotting outliers.

Worked Example: Bearing Life Test

Consider a bearing life test with 10 units. Six failed at 120, 180, 250, 310, 400, and 520 hours. Four units were still running when the test was suspended at 600 hours. We want to estimate the Weibull parameters and the B10 life.

Using MLE with the censored likelihood above, a numerical solver returns β ≈ 1.85 and η ≈ 410 hours. The B10 life — the time by which 10% of the population is expected to fail — is:

t_0.10 = η × [−ln(1 − 0.10)]^(1/β)
       = 410 × [0.10536]^(1/1.85)
       ≈ 410 × 0.301
       ≈ 123 hours

Because β > 1, the hazard rate is increasing, indicating wear-out. If the same data were fitted with an exponential model, the constant hazard assumption would underestimate early-life risk and overestimate late-life risk, leading to poor maintenance decisions.

Choosing Among the Three Distributions

In practice, the Weibull is the most versatile starting point. If β estimates close to 1 and the exponential fits adequately by likelihood ratio test, the simpler model may be preferred. If the data show a pronounced early peak in hazard, lognormal may fit better. The engineering context — failure mechanism, test conditions, and prior knowledge — should always inform the choice, not statistical fit alone.

Practical Guidance for Test Engineers

When planning a test, decide in advance how censoring will be handled. If units are expensive, a time-censored test with a fixed duration may be more practical than running until all units fail. The likelihood framework handles this naturally, but the analyst must record the exact censoring time for each unit — not just the number of survivors.

For small samples (fewer than 10 failures), MLE can be biased. Bias-corrected estimators or Bayesian methods with informative priors may be preferable. Probability plotting with median ranks remains a robust exploratory tool. Always report confidence intervals for the parameters and for derived quantities like B10 life; a point estimate alone hides the uncertainty that matters for engineering decisions.

Finally, remember that a distribution is a model, not a truth. The goal is not to find the "correct" distribution but to choose one that is adequate for the decision at hand — whether that is setting a warranty period, scheduling preventive maintenance, or qualifying a design change.