Poisson Process - Yousef's Notes

Can be used to model:

number of car accidents at a site.
location of users in a wireless network.
requests for individual documents on a web-server.
the outbreak of wars.
photons landing on a photodiode. Also forms the basis for spatial and spatio-temporal models (e.g. number of infected people throughout Spain evolving by day)

A counting process ${N(t), t \geq 0}$ is a Poisson process with rate $\lambda > 0$ if

$N(0) = 0$
the process has independent increments
the process has stationary increments and

$$ P(N(t+s) - N(s) = n) = P(N(t) = n) = \frac{e^{-\lambda t}(\lambda t)^n}{n!} $$ The number of events in an interval of width $t$ is distributed as a Poisson with parameter $\lambda t$. $$ E(N(t)) = \lambda t \quad \text{and} \quad V(N(t)) = \lambda t $$

#Inter-Arrival Times

Let $T_1$ be the time of the first event and $T_n$ be the time between the $(n-1)$st event and the $n$th event. These are called inter-arrival times.

$T_1, \ldots, T_n$ are independent and exponential with parameter $\lambda$.

$$ P(T_1 > t) = P(N(t) = 0) = e^{-\lambda t}, $$

which is the expression for the Exponential. For the other $T$’s, the argument is similar, also using the independent and stationary increments property.

#Waiting Times

The time of the $n$th event, $S_n$, is called the waiting time until the $n$-th event.

$S_n$ can be seen as $\sum_{i=1}^{n} T_n$, the sum of independent exponential distribution. This can be shown to be a Gamma distribution with parameters $n$ and $\lambda$. Its mean is $n/\lambda$ and its variance is $n/\lambda^2$.