In this section, we consider the time it takes for the first event to occur in a homogeneous Poisson process — a continuous random variable.

Exponential distribution

In a homogeneous Poisson process with rate \(\lambda\) events per unit time, the time until the first event, \(Y\), has a distribution called an exponential distribution,

\[ Y \;\; \sim \; \; \ExponDistn(\lambda) \]

with probability density function

\[ f(y) \;\; = \; \; \lambda\; e^{-\lambda y} \]

and cumulative distribution function

\[ F(y) \;\; = \; \; 1 - e^{-\lambda y} \]

(Proved in full version)

The diagram below shows the shapes of a few typical exponential distribution.

All exponential distributions have their highest probability density at \(x = 0\) and steadily decrease as \(x\) increases.