When events occur at an approximately constant rate, the waiting time until the next event is often modeled by an exponential distribution.

With rate $\lambda>0$, its density is

$$f(x)=\lambda e^{-\lambda x},\qquad x\ge0,$$

and zero for negative $x$.

Tail probabilities

The probability of waiting longer than $t$ is

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

Therefore the probability of an event occurring within time $t$ is

$$P(X\le t)=1-e^{-\lambda t}.$$

Mean waiting time

The expected waiting time is

$$E[X]=\frac1\lambda.$$

A larger rate therefore corresponds to a shorter typical waiting time.

Memoryless property

The exponential distribution has the distinctive property

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

After already waiting $s$ units of time, the remaining waiting-time distribution is the same as it was at the beginning.

This property is mathematically convenient but also a strong modeling assumption: it says the process does not become more or less likely to occur merely because we have already waited longer.