How to use it
- Choose whether you know the rate λ or the mean waiting time μ, and enter it.
- Enter x to get P(X ≤ x), P(X > x) and the density.
- Optionally add x₂ for the probability between two values, and a probability p for the quantile.
Formulas
For X ~ Exp(λ), x ≥ 0:
- P(X ≤ x) = 1 − e^(−λx), P(X > x) = e^(−λx)
- P(x₁ < X ≤ x₂) = e^(−λx₁) − e^(−λx₂)
- f(x) = λe^(−λx)
- Quantile x_p = −ln(1 − p)/λ; median ln 2/λ; mean 1/λ; variance 1/λ²
Worked example
Customers arrive at 0.5 per minute (mean gap 2 minutes). The chance that the next one arrives within 3 minutes is 1 − e^(−1.5) = 0.7769, so 0.2231 that you wait longer. The chance of a gap between 3 and 5 minutes is e^(−1.5) − e^(−2.5) = 0.141. Half of all gaps are shorter than the median 1.38629 minutes, and 90 % are shorter than 4.60517 minutes.
Tips and common mistakes
- Keep units consistent: a rate per hour needs x in hours.
- The rate and the mean are reciprocals; entering a mean of 2 as a rate of 2 gives very different answers.
- The model assumes a constant rate; parts that wear out or queues with rush hours need other distributions.