PSD of a Poisson process

Earlier we have started talking about the Poisson processes. In the few posts before the summer holidays we have driven our discussion towards waiting time paradox, which is an interesting phenomenon we encounter in our day-to-day lives. Here, on Physics of Risk we have an interest in colors of noise exhibited by variety of stochastic processes. Thus in the next few posts let us examine the power spectral density of the Poisson process.

401st post!

The post we have posted before the summer vacation was our 400th post! We haven't noticed this milestone at the time, but let us then celebrate 401st post by looking at the language statistics of our posts. Particularly, we can ask a question whether Zipf's law applies for our posts.

Technically this 402nd post, but it was written before the previous post. So let us still celebrate now :)

Zipf's law

Zipf's law is an empirical observation, that often in popularity (frequency, or size) tables the popularity decays as power-law function of rank:

\begin{equation} \text{popularity} \sim \frac{1}{\text{rank}^\alpha} . \end{equation}

With the dependence being close to the inverse law (i.e., \( \alpha \approx 1 \)).

So, will it hold?

It is September again!

Over the summer I (Aleksejus) was contacted by some people from MathWorks who have seen my rant (post) about Matlab. I got kindly asked if I could provide a more detailed feedback. Sadly (or happily?) I have already forgot most of my experience with Matlab. There are lots of memes about Matlab online, so I guess I am not the only one with similar experience. To be fair, I think one of the issues is that a lot of Matlab handbooks are quite dated. Not in a sense that there are no new handbooks, but in a sense that the authors of many of even newer handbooks are not that well familiar with newest best practices in Matlab.

Also just few days ago we got "rug pulled" by our local network admins. Physics of Risk site went down due to local network configuration changes. When we noticed the issue, and inquired about it, we got a reply a long the lines of "Sorry. Made some changes. Yesterday." Now the changes are temporarily reversed, but... we have already relocated Physics of Risk to GitHub Pages. As we already store our source code on GitHub (see here), so why not?

Otherwise, the summer this year was not that eventful. So, Physics of Risk will be back with a new post on Tuesday! And afterwards we will continue talking about the point processes.

Waiting time paradox

Last time we have introduced a concept of waiting time. We have assumed that some time has passed since the last event and wanted to how long do we have to wait until another event. The time since the last event was a thing we knew. Now let us consider a slightly different case - what happens when we do not know when the last event happened, how long do we have to wait until the next event?

In other words, imagine you just have come to a bus stop. If you know that the average bus interarrival time is \( 15 \) minutes, then how long would you expect to have to wait? Straightforward answer seems to be \( 7.5 \) minutes, but is it a correct answer?