Exponential Distribution Calculator
Calculate cumulative and survival probabilities for an exponential distribution.
Exponential distribution
The cumulative probability is 1 − e^(−λx), while the survival probability is e^(−λx). It is commonly used to model waiting times.
Understanding the Exponential Distribution
The exponential distribution is commonly used to model waiting time between independent events that occur at a constant average rate.
Exponential Example
If events occur at an average rate of 2 per hour, the calculator can estimate the probability that the waiting time is less than or greater than a chosen duration.
Model Assumptions
The standard exponential model assumes a constant event rate and the memoryless property. If the rate changes substantially, another model may fit better.
Frequently Asked Questions
Can waiting time be negative?
No. Exponential waiting times are nonnegative.
What does a higher rate mean?
It generally means shorter expected waiting times.
How is it related to Poisson?
Poisson models event counts while exponential models time between Poisson events.
Exponential Distribution in Practice
The exponential distribution is commonly used to model the waiting time between independent events that occur at a constant average rate. Its rate parameter is usually written as λ, and the mean waiting time is 1/λ when the standard parameterization is used.
This calculator can help with queueing examples, reliability exercises and event-arrival models. Confirm whether your input is a rate or a mean because the two are reciprocals. If λ = 0.5 events per hour, the mean waiting time is 2 hours. Keep time units consistent throughout the calculation.
The model has a memoryless property, meaning the future waiting-time distribution does not depend on how long you have already waited. Real processes with changing event rates, aging components or dependent arrivals may not satisfy this assumption, so the exponential distribution should only be used when it reasonably fits the process.
Methodology, Assumptions and Interpretation
For an exponential waiting-time model, P(T≤t)=1−e^(−λt). It assumes a constant event rate and the memoryless property; real waiting times with aging, schedules or changing hazards may not follow this distribution.