Exponential Distribution Calculator

What Is the Exponential Distribution?

The exponential distribution models waiting time between independent events that occur at a constant average rate. Examples include time between bus arrivals, customer visits, or machine failures. It is memoryless: the chance of waiting another minute does not depend on how long you have already waited. For related discrete waiting-time models, see the Negative Binomial Distribution Calculator or Binomial Probability Distribution Calculator.

Key Formulas

With rate parameter $a$ (events per time unit) and wait time $X$:

$$P(X \leq t) = 1 - e^{-at}, \quad P(X > t) = e^{-at}$$

$$\mu = \frac{1}{a}, \quad \text{median} = \frac{\ln(2)}{a}, \quad \sigma^2 = \frac{1}{a^2}$$

Example: Cafeteria Arrivals

Average 15 students per hour means $a = 15/60 = 0.25$ per minute. Probability of waiting at most 3 minutes: $P(X \leq 3) = 1 - e^{-0.25 \times 3} \approx 0.528$, or about 52.8%.

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