Calculadora de Distribuição Exponencial
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
F(x) = 1 − e^(−λx)
About this calculator
The Exponential Distribution Calculator computes the cumulative probability P(X ≤ x) for a continuous random variable following an exponential distribution. The formula used is F(x) = 1 − e^(−λx), where λ is the rate parameter (λ > 0) and x is the threshold value. This distribution models the time between events in a Poisson process, such as customer arrivals or equipment failures.
Usage is straightforward: enter the rate λ and the value x (time or distance). The calculator directly applies the formula, returning the probability that the random variable is less than or equal to x. For example, if λ = 0.5 events per hour and x = 2 hours, the probability that the next event occurs within 2 hours is 1 − e^(−0.5×2) ≈ 0.6321.
Use this calculator in scenarios such as: estimating the probability of equipment failure within a period, modeling waiting times in queues, or analyzing system reliability. It is particularly useful in engineering, actuarial science, and operations research. Remember that the exponential distribution assumes independent events and a constant rate.
Cautions: the rate λ must be positive; negative or zero values will cause errors. The value x must be non-negative (x ≥ 0), as the exponential distribution is defined only for x ≥ 0. Also, the exponential distribution is memoryless, meaning the probability of a future event does not depend on elapsed time.
Frequently asked questions
What does the parameter λ mean in the exponential distribution?
λ is the rate of event occurrence, i.e., the average number of events per unit time. For example, λ = 2 means on average 2 events occur per hour.
How do I interpret the result P(X ≤ x)?
It is the probability that the time until the next event is less than or equal to x. For instance, P(X ≤ 3) = 0.8 means an 80% chance the event occurs within 3 time units.
Can I use this calculator for negative x values?
No. The exponential distribution is defined only for x ≥ 0. Entering negative x will result in an error or invalid probability.
What is the relationship between exponential and Poisson distributions?
The exponential distribution models the time between events in a Poisson process. If the number of events per interval follows Poisson, the time between events follows exponential.
Does the calculator provide the probability density function?
No, this calculator only provides the cumulative probability (distribution function). For the density, use a specific PDF calculator.