Adventures in Probability

>my professor projected video of himself writing on a piece of paper before a very large auditorium, and that guy was left-handed, and so his hand would cover his notes for like the entire time and it was impossible to see what he was writing. I only figured out that this was why it was so unpleasant like halfway through the class.

So many of my college math classes had some version of this professor who took a fascinating subject like linear algebra, statistics or algorithms and made it into a slog. The fact that most stats is taught by getting students to just memorize random ideas rather than building up a holistic and intuitive view really is a travesty.

Also makes sense why so many people, even though they took stats in college, hav e such a poor understanding of probability.

> I think if I were in charge of presenting this material to students I'd do it by introducing the concept of memorylessness and by showing how good memorylessness is, how many wonderful things you can do with it. And then one day I'd be like, "well, it sure would be nice if we had any distributions like that!" and then whirl around with my piece of chalk to deliver the exciting news that we do. Exactly one, in fact.

Incidentally, this also goes for the determinant of a matrix. It's got a lot of neat and desirable properties, and it turns out to be the only thing that does. When it was finally taught to me this way, those weird algorithms we use to compute this seemingly-arbitrary number finally made sense. (And, in fact, this is the easiest way to prove that all those algorithms have to be computing the same seemingly-arbitrary number. Because the algorithms preserve the properties that define The Determinant, and The Determinant is the unique thing that preserves all of those properties, so must those algorithms all be computing The Determinant, no matter how different they might look.)

So I can vouch that this style of explanation really does work, at least for people like me.

Poisson processes are neat, they always end up working nicely in ways that many other distributions/processes very much don't.

Splitting a Poisson process into two lower rate processes is a neat trick. Even better is that you can do the same to convert a Poisson process into one with a variable rate, provided that rate is lower than the original (original may be variable as well).

And the fact that the partial sums of a bunch of exponential distributions results in the same distribution of values as picking Poisson(lambda * time) values uniformly at random is pure magic.

This was a lot of fun. By skipping over the formulaic details and proof and explaining the lay of the land, it makes a good starting point to explore further.

Either mathematically or just some python dice rolls.

Really good. Same ethos as fast.ai courses.

Finally that you can add tbe /3 and /5 to get a /8 distribution makes intuitive semsw to me. / means lambda.

This is because if you have people arriving to a train station you could split them by eye colour and there is no reason a particular eye colour cause there to be dependencies. (Assuming spherical cows: families arriving excepted. Assume it is downtown rush hour)

I doubt that this model of a queue and the processing of its items by overlaying two independent poisson processes is statistically valid (one for items arriving, the other for processing those items). The processing starts only after the respective item arrived in the queue - so it's not independent - and this needs to be modeled accordingly or it requires a proof that this is equivalent to the suggested overlaying approach - that wouldn't be obvious or trivial.

I can't help but wonder if real systems have additional (perhaps subtle) signals, which can be provided to a neural network, which then outperforms these simple algorithms.

For example, customers arrive at the grocery store in clusters due to traffic lights, schools getting out, etc. Even without direct signals, a NN could potentially pickup on these "rules" given other inputs, e.g. time of day, weather, etc.

?

You can't combine rate of arriving with rate of leaving, can you? Leaving is dependent on arriving, so the latter distribution is dependent on the former.

How does one know the application of such a Math concept for a particular software problem? I couldn't guess in a million years.

hey this might be a stupid detail in implementation, but if the poisson simulation of arrival and completion are started cold, then random assignment of events could assign a completion before an arrival, or there could be more completions than arrivals? Arrivals should always be greater or equal to completions?

Edit: I remembered by stats class. merging poisson distributions requires Independent processes. since start and completion are dependent, its not mathematically valid.

but I am an engineer and it works, so whatever

my Msc thesis in 2004 was about this: a probabilistic model for last-recently used queues, based on poisson processes, for network packets flows. The model worked OK on a couple of datasets I could get at that time. I tried to publish the work but got rejected a couple of times, then I gave up. If anyone wants to read it (even try to publish) I am happy to share it.