Modeling With the Exponential Distribution

Many real-world waiting times — the time until the next call at a help desk, until a radioactive atom decays, or until the next bus arrives — are modeled by the exponential distribution. If events occur randomly at an average rate of $\lambda$ events per unit of time, then the waiting time $X$ until the next event satisfies

$$X \sim \text{Exp}(\lambda),$$

with cumulative distribution function

$$P(X \le t) = 1 - e^{-\lambda t}, \qquad t \ge 0,$$

and survival function

$$P(X > t) = e^{-\lambda t}, \qquad t \ge 0.$$

The mean waiting time is $E[X] = \dfrac{1}{\lambda}$.

When a problem describes a mean waiting time $\mu$ instead of a rate, set $\lambda = \dfrac{1}{\mu}$ before applying the formulas. Always make sure $\lambda$ and $t$ are expressed in compatible units.

For example, suppose calls arrive at a help desk at a rate of $\lambda = 4$ calls per hour. The probability we wait more than $30$ minutes (so $t = 0.5$ hours) for the next call is

$$P(X > 0.5) = e^{-4 \cdot 0.5} = e^{-2} \approx 0.135.$$