Enter model inputs
Use the exponential model for constant hazard. Use the Weibull model when hazard changes with age.
Example data table
Example values below use a Weibull model with β = 1.8 and η = 500 hours.
| Time | Hazard | Survival | Failure CDF | Cumulative Hazard |
|---|---|---|---|---|
| 100 hours | 0.000993 | 0.946306 | 0.053694 | 0.055189 |
| 250 hours | 0.002068 | 0.750381 | 0.249619 | 0.287175 |
| 400 hours | 0.003011 | 0.512113 | 0.487887 | 0.669209 |
| 550 hours | 0.003885 | 0.305089 | 0.694911 | 1.187153 |
| 700 hours | 0.004712 | 0.160022 | 0.839978 | 1.832443 |
Formula used
Exponential model
Hazard: h(t) = λ
Cumulative hazard: H(t) = λt
Survival: S(t) = e-λt
Failure probability by time t: F(t) = 1 - S(t)
Density: f(t) = λe-λt
Weibull model
Hazard: h(t) = (β / η)(t / η)β - 1
Cumulative hazard: H(t) = (t / η)β
Survival: S(t) = e-(t / η)β
Failure probability by time t: F(t) = 1 - S(t)
Density: f(t) = h(t) × S(t)
Interval probability
Unconditional interval failure probability: P(t0 < T ≤ t1) = S(t0) - S(t1)
Conditional interval failure probability: P(T ≤ t1 | T > t0) = 1 - S(t1) / S(t0)
Expected failures among n surviving items: n × conditional interval probability
How to use this calculator
- Select either the Weibull or exponential lifetime model.
- Enter the interval start time and interval end time.
- Provide λ for exponential, or β and η for Weibull.
- Enter the number of items currently exposed to risk.
- Choose a readable unit label such as hours or cycles.
- Press the calculate button to view the summary cards.
- Review the Plotly graph for time-based risk behavior.
- Export the result summary to CSV or PDF when needed.
Frequently asked questions
1. What does the hazard rate measure?
It measures the instantaneous failure intensity at a given time, assuming the item has survived up to that moment.
2. When should I use the exponential model?
Use it when failures occur with a constant hazard over time. It is common for memoryless lifetime assumptions.
3. When is the Weibull model better?
Use Weibull when hazard changes with age. It can model infant mortality, random failures, and wear-out behavior.
4. What does β mean in the Weibull model?
β controls the hazard trend. Values below one decrease hazard, one gives constant hazard, and values above one increase hazard.
5. What does η represent?
η is the Weibull scale parameter. It stretches the time axis and strongly influences when most failures begin appearing.
6. Why are both conditional and unconditional probabilities shown?
Unconditional probability uses the full lifetime origin. Conditional probability focuses on items that already survived to the interval start.
7. Why can the hazard at time zero be infinite?
For Weibull models with β below one, early-life failures dominate. That shape creates very high initial hazard near zero.
8. What does expected failures mean here?
It estimates how many items may fail in the chosen interval, assuming the entered number of items is currently surviving.