Smooth computable functions with "chaotic" walks

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D-503
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Smooth computable functions with "chaotic" walks

Postby D-503 » Wed Nov 13, 2013 6:13 am UTC

I started with a question on Stack Overflow about generating deterministic random events:
http://stackoverflow.com/questions/1962 ... on-process

One I realized my question was more calculus then programming I started this question on Math Overflow:
http://math.stackexchange.com/questions ... istributed

However, my Math Overflow question never received any answers, even after I placed a bounty on it, so I though I'd try bringing it to the xkcd forum to discuss.

Summarized, my Math Overflow question is how do I construct a function I can use to compute the number of deterministically simulated Poisson events between two points in time. (EDITED to address the issues brought to light by jestingrabbit)

I will also try to give an intuitive problem definition here based on my current thoughts about it. At it's core, I think the challenge is to come up with a mathematical function that looks like a up-and-down scribble above 0 that can be deterministically computed and integrated. This is where the thread's title comes from. If I had such a function, I would use it's integral to represent the number of events that occurred since time0. Since I would want those events to be Poisson distributed on any given interval, there may be additional constraints on the random up and down function. For example, it would be able to reach unlimited height, but only rarely and briefly, before falling back towards zero.

Thanks for reading! I hope this is clear enough to understand, please ask questions if not.
Last edited by D-503 on Wed Nov 13, 2013 8:21 am UTC, edited 2 times in total.

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jestingrabbit
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Re: Smooth computable functions with "chaotic" walks

Postby jestingrabbit » Wed Nov 13, 2013 6:17 am UTC

At one point there you've got little f taking two arguments, then later you are integrating it over an interval. What is the domain and range of f?
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D-503
Posts: 84
Joined: Sun Apr 15, 2012 11:35 pm UTC

Re: Smooth computable functions with "chaotic" walks

Postby D-503 » Wed Nov 13, 2013 7:49 am UTC

jestingrabbit wrote:At one point there you've got little f taking two arguments, then later you are integrating it over an interval. What is the domain and range of f?


f is meant to be a function that returns the number of events that happens between two points in time. Sorry, I think I messed up proposition 2, I think it should be that the PDF of f for a random t1,t2 pair is the Poisson PDF where the interval length is t2 - t1. Also, it's possible to use a single variable function g(t) = f(0, t) because f(t1,t2) = g(t2) - g(t1). The domain and range is the positive reals.


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