MathCalcLab

Exponential Distribution Calculator

Calculate exponential distribution probabilities with PDF and CDF for modeling time between events

The rate parameter λ > 0 represents the average number of events per unit time

Optional: Enter a time value to calculate PDF and CDF

Results

Enter values and click Calculate to see the result.

Theory & Formula

Theory

The exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate λ. It is widely used in reliability engineering, queuing theory, and survival analysis.

Probability Density Function (PDF)

\(f(x; \lambda) = \begin{cases} \lambda e^{-\lambda x} & \text{for } x \geq 0 \\ 0 & \text{for } x < 0 \end{cases}\)

Cumulative Distribution Function (CDF)

\(F(x; \lambda) = \begin{cases} 1 - e^{-\lambda x} & \text{for } x \geq 0 \\ 0 & \text{for } x < 0 \end{cases}\)

Properties

\(\text{Mean: } \mu = \frac{1}{\lambda}\)\(\text{Variance: } \sigma^2 = \frac{1}{\lambda^2}\)\(\text{Standard Deviation: } \sigma = \frac{1}{\lambda}\)\(\text{Median: } \frac{\ln(2)}{\lambda}\)

Memoryless Property

The exponential distribution has a unique memoryless property: the probability of an event occurring in the next t time units is independent of how much time has already elapsed.

\(P(X > s + t \mid X > s) = P(X > t)\)

Example

If customers arrive at a rate of λ = 0.5 per minute, the average time between arrivals is 1/0.5 = 2 minutes. The probability that the next customer arrives within 3 minutes is F(3) = 1 - e^(-0.5×3) ≈ 0.777 or 77.7%

Related Calculators

Poisson Distribution Calculator

Calculate Poisson probabilities for rare events

Uniform Distribution Calculator

Calculate uniform distribution probabilities

Normal Distribution Calculator

Calculate probabilities for normal distribution

Exponential Distribution Calculator | PDF & CDF | MathCalcLab | MathCalcLab