The Poisson counted events per interval. Flip it around to ask about the gap between events, the time until the next failure, and you get the exponential distribution, the continuous twin of the geometric and the workhorse of reliability engineering.
hardware_component_lifespans.csv records lifespan_years for 1,000 components (plus
type). We model time-to-failure as exponential and ask about reliability, warranties, and aging.
Fit the Failure Rate
The rate is one over the mean lifespan: λ = 0.20 per year, a mean lifetime of 5 years. The exponential is right-skewed, many parts fail relatively early while a few last a long time, and its mean and standard deviation are equal (both 5 years), a defining signature.
The median, ln(2)/λ ≈ 3.4 years, sits well below the mean, the telltale gap of a right-skewed distribution. Half the parts are gone by 3.4 years even though the average lifetime is 5.
A matching curve is not proof. The companion notebook validates the fit with an exponential QQ plot (the points land on the diagonal) and a formal Kolmogorov-Smirnov test (D ≈ 0.02, p ≈ 0.94, so we cannot reject the exponential). The model is earned, not assumed, the discipline every distribution fit deserves.
Reliability from the CDF
Reliability is a CDF question. The chance a part fails before age x is 1 − e−λx; its complement, the survival function e−λx, is the chance it is still running.
| Age | P(failed by then) | survival (still alive) |
|---|---|---|
| 1 year | 0.182 | 0.818 |
| 2 years | 0.331 | 0.669 |
| 5 years (one mean) | 0.634 | 0.366 |
| 10 years | 0.866 | 0.134 |
About 18% of components fail within the first year (the early-fatigue risk, matched by the data), and 37% survive to 5 years, exactly e−1 since one mean lifetime has elapsed. To cap early failures at 5%, the warranty should run only about 0.26 years: the CDF turns a risk target straight into a policy.
Memoryless: No Wear-Out
The exponential shares the geometric's most surprising trait: it is memoryless. A part that has run for five years has the same chance of lasting one more as a brand-new one, its hazard rate is constant. Parts modeled this way simply do not age.
In the simulation, P(survive one more year) is 82% whether the part is fresh or already five years old, memorylessness in action. That constant hazard is the exponential's great convenience, but also its blind spot: real components that genuinely wear out have a rising hazard and need the Weibull distribution. Knowing when "as good as new" is realistic, and when it is not, is the engineer's judgment.
Run the reliability analysis
The companion notebook fits the exponential failure rate, overlays the density on the lifespan histogram, tabulates failure and survival probabilities from the CDF, derives a warranty length from a risk target, and demonstrates the memoryless, constant-hazard property by simulation.
View opens the rendered notebook instantly (no setup). Open in Colab runs &
edits it live in your browser. To run locally, install numpy, pandas, scipy,
and matplotlib and launch jupyter notebook.
🎓 Key Takeaways
- ✓Time-to-failure is exponential: PDF λe−λx, mean 1/λ; here λ = 0.20/yr, mean 5 years.
- ✓Right-skewed: mean = std = 5, but the median is only 3.4 years.
- ✓Reliability is the CDF: P(fail by x) = 1 − e−λx; 18% fail in year one, 37% survive to 5 years.
- ✓Invert it for policy: a 5%-early-failure warranty runs about 0.26 years.
- ✓Memoryless / constant hazard: a 5-year-old part is as good as new; genuine wear-out needs a Weibull.
Quiz: Test Yourself
Eight quick questions on the exponential case study. Answer them, hit Check Answers, and keep refining until you score 100%. Your progress is saved, so you can hop back to the chapter and return anytime.