Is there any known probability distribution that behaves like a rock climber doing lead belay climbing (where they're clipping the rope in at various points as the climb up a rock) - where there are usually small gains (like how a rock climber usually makes slow, steady progress up a rock), but less frequently, there are really big drops (like when a rock climber falls)? So maybe it would look something like this:

rock climbing distribution

(So usually things go pretty good, but when they go bad, they are really bad. This would also be a good model for the modeling markets prone to developing bubbles).

I suspect that such a "distribution" would be a good model for things like the "capacity utilization" in this graph:

capacity utilization graph

(Note how typically, there are small gains, but periodically, there are big drops)

So my question is: is there a known distribution that would model things like this already?

The poisson reminds me of it, but is a bit different. Perhaps, the answer is a model which is 2 separate normal distributions - one for normal small gains and one for massive losses (and of course, some probability distribution that defines which of the two is drawn from in a given sample).


Recently, I was modeling some mutual funds with a Normal distribution, and a thought occurred to me: digesting historic returns of a mutual fund and outputting 2 parameters, mean and standard deviation, is a pretty big information loss. Obviously, trying to build a model is a "lossy" "compression" technique, but I'm concerned there could be some valuable information being lost in a Normal Distribution model we could maintain in a better model like the one I describe above.


You seem to have probability distribution functions (PDFs) confused with time series. A PDF has integral 1, so it can't have arbitrarily high bumps arbitrarily far along the real line. A time series is typically modeled not as a single random variable (and hence a single distribution) but as a function mapping from time to random variables; that is, a random process.

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    $\begingroup$ I disagree about the confusion. Given bounded falls, bounded distances out the real line, I can easily imagine a proper PDF for this, but I do not know what this would be. I could simulate it given the probability that the next move was a fall, or a rise, and some distribution for their respective magnitudes. There is lots of work on mixture distributions, but I know little more than that. $\endgroup$ – astaines Sep 27 '16 at 10:07
  • $\begingroup$ Sorry, I should have labeled my axes, but the idea is this is a PDF, with an integral of 1, representing the probability of getting each specific return rate (in this case, as a percentage). $\endgroup$ – caleb Sep 27 '16 at 14:00
  • $\begingroup$ @caleb I see. In that case, you probably want to construct a mixture distribution, such as a mixture of two normal distributions. $\endgroup$ – Kodiologist Sep 27 '16 at 15:34

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