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.