Exponential Distribution

Continuous waiting times: tail, mean, and memorylessness.

The idea

The exponential distribution models a continuous waiting time: the time until an atom decays, a customer arrives, a component fails. A nonnegative random variable $X$ is exponential with rate $\lambda > 0$ when its density is $f(x) = \lambda e^{-\lambda x}, \qquad x \ge 0.$ The rate counts events per unit time, and the mean wait is its reciprocal, $\mathbb{E}[X] = 1/\lambda$: doubling the rate halves the typical wait. Integrating the density from $t$ onward gives the tail probability $\mathbb{P}(X > t) = e^{-\lambda t}$, the probability of still waiting at time $t$, and its complement $\mathbb{P}(X \le t) = 1 - e^{-\lambda t}$ is the cumulative distribution.

The defining property is memorylessness. Because $e^{-\lambda(s + t)} = e^{-\lambda s} e^{-\lambda t}$, the tail satisfies $\mathbb{P}(X > s + t \mid X > s) = e^{-\lambda t}$: given that the wait has already lasted to time $s$, the remaining wait has exactly the original distribution. The exponential is the only continuous distribution with this property, and the property limits the model: a component that wears out with age should not be described by it.

Ways to work on it

• Walkthrough. Density, tail and cumulative, the mean, and the memoryless property.
• Practice. Compute a tail or cumulative probability for a given rate.
• Hardest. Recover the rate from a probability, then apply memorylessness.

Not sure where to start? Take the ten-question placement test.