Translate a constant rate into a waiting-time probability

Define the task

Define the event, the time origin and the unit before selecting a waiting-time distribution. A constant-rate model assigns the same instantaneous event rate throughout follow-up among units still waiting. Write down why that assumption could be reasonable for the synthetic process under study. An average count per hour alone does not establish it.

Carry out the check

Under an exponential model, survival to time t is exp(−rate × t). Subtract survival from one to obtain the probability of an event by that time. Keep rate and time in reciprocal units. For very small products, evaluate the event probability with an expm1-based calculation to reduce cancellation. Record the rate as an assumption or describe its estimation population.

Worked hypothetical example

For a hypothetical simulator with rate 0.2 events per hour, mean waiting time is five hours. The probability of an event within three hours is 1−exp(−0.6), about 0.4512. Waiting another three hours conditional on no event so far gives the same probability under this model. A changing rate, competing events or heterogeneous processes can invalidate that memoryless prediction.

Method checked 11 November 2026 against R documentation: The Exponential Distribution.

Open the exponential waiting probability.

Download the waiting-time assumption record.