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Comment by atleastoptimal | Hacker News Reader

There's a subset of this I think about sometimes, not sure if it's actually a thing or been looked at/is valid.

Let's say I'm waiting for a package to arrive and it's late. As each minute passes, does the chance of it coming the very next minute increase or decrease?

One could say it increases because since it will eventually come and I have passed all the previous minutes they haven't come, then the remaining minutes until it comes comprise a smaller percentage of the all the remaining minutes, increasing the chance each minute.

However up to a certain point the chance may decrease, as in, if it is "late" in a substantial sense, whatever caused it being something that delayed it a significant amount of time increases as time elapses, as the chances of it being something small that caused the lateness decreases as each minute passes.

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And you can model this through two survival functions, one thin-tailed and one heavy-tailed. As time increases, whatever your initial probability of "package will be very late" it will be dwarfed by the posterior after evidence of a long waiting time.

It's the exact same technique used in https://two-wrongs.com/forecasting-covid-19-variants.html except applied to "packages that may be late" instead.

> it will eventually come

It might also never come.

Sometimes packages just disappear in transit.

Maybe it lost its label.

Maybe it got broken.

Maybe someone stole it.

Etc.

My experience with packages goes like:

- Most often things I order in my own country or from a neighbouring country arrive exactly on the expected day, or at least the next business day after that. This makes sense because shorter travel has less opportunity for things to go wrong.

- Things from far away have their estimate set for like 60 days from now, because no one knows what boat it will end up etc. Sometimes these things show up quickly. Most of the time it takes a while, but still it arrives before the long estimate date. Sometimes they show up a bit late.

- If they are very delayed then in both cases it means the package is just gone and will never arrive. So I inform the platform that the package was lost in transit and they refund the purchase.

This is a sort of transformation of the bathtub curve.

Replace it with waiting for a bus and you have a classic textbook example for teaching Poisson process!

https://en.wikipedia.org/wiki/Poisson_point_process

https://stats.stackexchange.com/questions/122722/please-expl...

It's specifically not a Poisson process, which assumes that the chance of a bus arriving at any given time is constant.

A bus arrival has some kind of multimodal distribution, where the bus either comes on time or very late, with some noise in each case.

The link between the Poisson process and the distribution of bus arrival times, is firstly that it's a good idealized example for understanding and reasoning about Poisson and exponential distribution, and secondly that the Poisson process somehow seems to be an accurate representation of our mental model of how waiting for a bus works.

If buses are scheduled to come quite frequently (and don't disappear, so they do arrive with that frequency on average), but the variance in their arrival times is a lot bigger than the average gap between them, then the distribution is actually quite close to Poisson.

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