The time until the first event in a homogeneous Poisson process has an exponential distribution. We next find the distribution of the time until the second event in the process.
Time until second event
If \(X\) is the time until the second event in a Poisson process with rate \(\lambda\), it has probability density function
\[ f(x) \;\;=\;\; \begin{cases} \lambda^2 x e^{-\lambda x} & x \gt 0 \\[0.3em] 0 & \text{otherwise} \end{cases} \](Proved in full version)
The pdf of the time until the third event in the Poisson process can be found in a similar way.
Time until third event
If \(X\) is the time until the third event in a Poisson process with rate \(\lambda\), it has probability density function
\[ f(x) \;\;=\;\; \begin{cases} \dfrac{\lambda^3}{2!} x^2 e^{-\lambda x} & x \gt 0 \\[0.5em] 0 & \text{otherwise} \end{cases} \](Proved in full version)
We now generalise this to the time until the \(k\)'th event.
Time until k'th event
The time until the \(k\)'th event in a Poisson process with rate \(\lambda\), has a distribution called an Erlang distribution
\[ X \;\; \sim \; \; \ErlangDistn(k, \lambda) \]with probability density function
\[ f(x) \;\;=\;\; \begin{cases} \dfrac{\lambda^k}{(k-1)!} x^{k-1} e^{-\lambda x} & x \gt 0 \\[0.5em] 0 & \text{otherwise} \end{cases} \](Proved in full version)