Hacker News new | ask | show | jobs
by theoh 2793 days ago
It is impossible to catch more than one inbound bus on any given occasion, whereas any number of outbound buses might pass.

BTW, on a slightly unrelated point, if there's no timetable, but the interval between buses is maintained reliably, the expected waiting time is uniformly distributed over that interval.

If you have to get a second bus, you need to convolve two of those two uniform distributions to find out the distribution of overall journey times. This is a trapezoidal distribution, which is just about analytically manageable.

But a journey with two transfers (3 buses in total) results in a likely overall time distributed according to a uniform distribution convolved with a trapezoidal distribution, which is a very weird non-smooth shape. You can see why people choose to model distributions with Gaussians, which are well-behaved (convolve two Gaussians, get another Gaussian). The Gaussian just lends itself ideally to recursive applications, hence recursive filtering (e.g. Kalman filters).
1 comments
LolWolf 2793 days ago
Also, gaussians are great approximations for large n, too, since the convolution of any distribution with itself n times (for n "large enough") is close to gaussian (by the CLT. More generally, there are very nice error estimates for many distributions).

I suspect this analysis can be carried out and yield quite good results in the gaussian case (a careful analysis might even yield error bounds on the result).

link
theoh 2793 days ago
Yes. If you spend your whole life on one long multi-transfer bus journey, you'll end up with a gaussian.

It's a bit less clear that gaussians should be used when e.g. fitting a coordinate to an astronomical feature, which might not actually be symmetrical.

The other useful property that the gaussian has is its separability, in the 2D case. That is unique to the gaussian and counts for a lot.

link
LolWolf 2793 days ago
Eh, I don’t think that many are required. Convergence to a Gaussian is pretty fast (you should check out page 299 of [0]), at four or five a Gaussian is already a quite good approximations.

———

[0] https://www.dartmouth.edu/~chance/teaching_aids/books_articl...

link