Direct answer
Exponential models a constant hazard, Weibull permits monotone hazard shapes, gamma can represent accumulated waiting stages, and lognormal models a positive quantity whose logarithm is normal.
Visual explanation
Positive-duration models imply different survival shapes
What this calculation tells you
Time-to-event distributions describe how probability accumulates across positive durations. Density, survival, and hazard functions answer different questions about timing.
Choose a candidate from the physical or operational process, inspect support and hazard implications, and use diagnostics that respect censoring and sampling—not only a histogram of completed times.
Where it is used
Reliability
Compare entered lifetime-model scenarios without approving a component design.
Queues
Model waiting or interarrival durations under stated assumptions.
Operations
Explore completion-time distributions and service reliability.
Education
Connect positive distributions with survival and hazard views.
Common situations
- Comparing constant and changing hazard assumptions.
- Choosing a positive-duration candidate model.
- Reading survival rather than density.
- Recognizing when censoring needs specialist analysis.
Start with the statistical question
Choose a candidate from the physical or operational process, inspect support and hazard implications, and use diagnostics that respect censoring and sampling—not only a histogram of completed times.
Four survival curves share a panel while companion hazard sketches reveal constant, increasing, decreasing, and non-monotone possibilities that density alone can hide.
Worked example
An exponential model with rate λ has survival exp(−λt) and constant hazard λ. A Weibull shape above 1 implies an increasing model hazard, while shape below 1 implies decreasing hazard.
Assumptions that carry the result
Simple calculators assume fully observed independent values and a fixed parameterization. Real reliability and survival analyses may involve censoring, truncation, repair, competing risks, covariates, and changing environments.
Interpret the result without overreaching
A fitted curve does not prove failure physics or predict an individual lifetime. Engineering and clinical decisions require domain-specific validation and uncertainty analysis.
- Choosing a model solely by the mean.
- Ignoring censoring when only completed events are plotted.
- Confusing density, probability, survival, and hazard.
Practical questions
Frequently asked questions
Is Weibull always better than exponential?
No. It is more flexible, but flexibility does not guarantee a better or more interpretable model.
Can lognormal values be negative?
No. The modeled variable is positive; its logarithm is normal.
What does constant hazard mean?
Under the model, instantaneous event propensity conditional on survival does not depend on elapsed age.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.