Poisson process: Interarrival times

May 16, 2023 Aleksejus Kononovicius #Interactive models #statistics #statistical physics #Poisson process

Last time we have built a quick model for independent student arrival. We have assumed that there is some probability that a student will arrive during some short time interval. Then we have taken the continuum limit, and forgot the microscopic model. In this post let us take a look what was left behind the scenes in the last: interarrival time distribution.

Geometric distribution

If we care not about the number of successful Bernoulli experiments, but about number of Bernoulli trials before the next successful one, then we are concerned not with the binomial distribution, but with the geometric distribution, whose probability mass function is given by

\begin{equation} P(X=k) = (1-p)^{k-1} p . \end{equation}

It looks very similar to the probability mass function of the binomial distribution, but for a single success. Also it lacks binomial coefficient. This is because there is just one way to have \( k - 1 \) failed trials before having a single successful trial at \( k \)-th attempt.

Survival function (probability that nothing has happened until step \( k \)) of the geometric distribution is given by:

\begin{equation} P(X > k) = (1-p)^k . \end{equation}

Exponential distribution

If we take continuum limit of this cumulative distribution function, with \( p = \lambda \Delta t \) and \( k = t / \Delta t \):

\begin{equation} P(T > t) = \lim_{\Delta t \rightarrow 0} \left( 1-\lambda \Delta t \right)^{\frac{t}{\Delta t}} = \exp(-\lambda t). \end{equation}

We have obtained the survival function of the exponential distribution. Therefore instead of doing rejection-based sampling (checking whether a student has arrived for each specific discrete time step), we can simply generate continuous interarrival times from the exponential distribution. This is actually what is happening in the app from the previous post.

Interactive app

This app, unlike the previous one, uses rejection-based sampling. Namely, it considers discrete time steps (with width \( \Delta t \)), and for each of the time steps it check whether the student arrives (event probability is given by \( \lambda \Delta t \)). Simulated probability density function is shown by the red curve, while the black curve corresponds to the exponential distribution.

Note that the theoretical curve fits only when \( \lambda \Delta t \) is reasonably small.