Solve reliability and waiting-time questions in one place. Switch between rate, scale, percentiles, and intervals. Review formulas, sample values, and export results without clutter.
| Example Rate λ | Time x | PDF f(x) | CDF F(x) | Survival S(x) |
|---|---|---|---|---|
| 1.2 | 0.5 | 0.658574 | 0.451188 | 0.548812 |
| 1.2 | 1.0 | 0.361433 | 0.698806 | 0.301194 |
| 1.2 | 2.0 | 0.108862 | 0.909282 | 0.090718 |
The exponential model uses a positive rate, written as λ.
Probability density function: f(x) = λe-λx, for x ≥ 0.
Cumulative distribution function: F(x) = 1 - e-λx, for x ≥ 0.
Survival function: S(x) = e-λx.
Interval probability: P(a ≤ X ≤ b) = F(b) - F(a).
Quantile function: Q(p) = -ln(1-p) / λ.
Mean: 1 / λ. Variance: 1 / λ². Standard deviation: 1 / λ.
Hazard rate: h(x) = λ. The hazard stays constant.
Memoryless tail rule: P(X > x+s | X > x) = e-λs.
The exponential distribution calculator helps you study waiting times between random events. It fits processes with a constant event rate. Examples include machine failure timing, call arrivals, and service delays. This page computes density, cumulative probability, survival chance, interval probability, and percentiles. It also reports mean, variance, standard deviation, entropy, hazard rate, and memoryless tail results. These outputs help analysts compare risk, reliability, and response speed from one form.
The exponential model is widely used in probability, reliability engineering, operations research, and queue analysis. It works when events happen independently and the average rate stays stable. The distribution is simple, but it answers practical questions fast. You can estimate how long a system may run before failure. You can measure the chance that a task finishes within a target time. You can also test whether a long wait changes the future chance of another wait.
PDF values describe local likelihood around one time point. CDF values show the chance that an event happens by time x. Survival values show the chance that the event has not happened yet. Interval probability helps when you need a range instead of one point. Quantiles help set service thresholds, maintenance limits, and service level targets. The hazard rate stays constant in this model. That constant hazard is the key feature behind the memoryless property.
Use rate λ when you know events per unit time. Use scale θ when you know average waiting time. Smaller mean values imply faster events. Larger rate values also imply faster events. Review the generated table to compare several times at once. Export the summary when you need a report, worksheet, or audit record. This makes the calculator useful for teaching, forecasting, process design, and reliability reviews.
It measures the waiting time until the next event in a process with a constant average rate. It is common in reliability studies, arrivals, failures, and queue timing work.
Enter rate λ when you know events per unit time. Enter scale θ when you know the average waiting time. They are reciprocals of each other.
PDF shows the density near one time point. CDF shows the probability that the event occurs on or before a chosen time.
The exponential model assumes the event rate does not change over time. Because of that assumption, the hazard remains equal to λ for all valid times.
It means past waiting does not change future waiting behavior. If the model fits, the chance of waiting s more units stays the same, no matter how long you already waited.
No. Exponential waiting times begin at zero. Negative time values do not represent valid observations for this distribution.
The percentile or quantile tells you the time below which a chosen share of outcomes falls. For example, the 90th percentile covers 90 percent of cases.
Exports help you save calculations, share analysis, document assumptions, and reuse the table in reports, audits, class notes, and planning files.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.