How to use
Use this calculator for a waiting time until the next event when occurrences have a constant rate and independent increments. Enter a positive rate λ and a time x to find the chance that the wait is no longer than x, or longer than x. Choose the highlighted event and press Calculate. For a target cumulative probability, select Probability → time and enter q instead of x. q=0.9 means the 90th percentile, regardless of which side you highlight.
Understand the probabilities
For x≥0 the cumulative probability is P(X≤x)=1−exp(−λx), and the upper probability is P(X>x)=exp(−λx). The mean waiting time is 1/λ and the variance is 1/λ². In quantile mode the time is x=−ln(1−q)/λ. The shaded area under the density curve represents the selected probability. The curve height λ exp(−λx) has units of inverse time and may exceed one without any probability exceeding 100%.
Worked examples
With a rate of 2 events per hour and x=0.5 hours, the probability of waiting at most half an hour is 63.21%; the probability of waiting longer is 36.79%. The mean is 0.5 hours and the variance is 0.25 hours squared. Enter 0.5, not 30, when the rate is measured per hour.
With rate 2 and quantile q=0.9, the required time is about 1.15 hours. Using the unrounded time in the cumulative formula gives 90%. At q=0 the time is zero. At q=1 the result is infinity: no finite waiting time covers the whole distribution. Infinity is a boundary result, not an input error or a promise that an event will never occur.
Uses and limits
The rate is supported from 0.000000001 through 1,000,000,000 per time unit. Direct time inputs range from zero through 1,000,000,000,000; quantile inputs range from zero through one. Decimal point or decimal comma is accepted without thousands separators. Use the same time unit throughout. The tool does not infer a rate or convert units automatically. It is a model for constant-rate waiting, not a forecast of scheduled or aging processes.
Frequently asked questions
Why is P(X=x) not shown?
A continuous waiting time has zero probability at any single exact time. Probabilities belong to intervals and areas. The density at a point is useful for drawing the curve but is not the chance of observing that exact time.
Why does the curve stop?
The graph displays zero through six mean waiting times. The remaining right tail is about 0.25% and is explicitly reported. A selected boundary may be beyond the graph, including infinity. The displayed numeric probability always includes the full selected tail, not only the visible shaded area.
What does the memoryless assumption mean?
After already waiting a while, the model gives the same probability of an additional wait as at the start. This can be useful for random arrivals, but it is inappropriate when an item wears out or an appointment becomes due. Verify that a constant rate is sensible for your use.
How are results rounded and copied?
Ordinary results show at most two decimal places; very small positive values use scientific notation, and positive probabilities below numerical resolution are marked < 1E−300. Copy result includes the confirmed rate, mode, input and results. Editing a field or selection invalidates the numbers, drawing and Copy state together; Clear empties the numeric fields.