Exponential Distribution Calculator — Formula
## How to Calculate Exponential Distribution Probabilities
### Formulas
**PDF: f(x) = lambda * e^(-lambda*x)** for x >= 0
**CDF: P(X <= x) = 1 - e^(-lambda*x)**
The exponential distribution models the time between events in a Poisson process. The parameter lambda is the rate (events per unit time). The mean waiting time is 1/lambda. It has the memoryless property: P(X > s+t | X > s) = P(X > t).
### Formulas
**PDF: f(x) = lambda * e^(-lambda*x)** for x >= 0
**CDF: P(X <= x) = 1 - e^(-lambda*x)**
The exponential distribution models the time between events in a Poisson process. The parameter lambda is the rate (events per unit time). The mean waiting time is 1/lambda. It has the memoryless property: P(X > s+t | X > s) = P(X > t).
Esempio Risolto
Buses arrive at a rate of 0.5 per minute. What is the probability of waiting at most 3 minutes?
- lambda = 0.5, x = 3
- PDF: f(3) = 0.5 * e^(-0.5*3) = 0.5 * e^(-1.5) = 0.5 * 0.2231 = 0.1116
- CDF: P(X <= 3) = 1 - e^(-1.5) = 1 - 0.2231 = 0.7769
- P(wait > 3 min) = e^(-1.5) = 0.2231
- Mean waiting time = 1 / 0.5 = 2 minutes