Quincunx for non-Gaussian distributions?

July 31, 2015 10:43 AM

Galton's quincunx machine was a machine to demonstrate the bell curve. Are there any such machines for non-Gaussian probability distributions?

Specifically, lognormal, Weibull, power law distributions would be appreciated. I would be very amused if there was such a thing for a Cauchy distribution also. Or an equivalent decision tree (given that the quincunx itself is obviously a stochastic decision tree for each ball that falls in)

posted by curuinor to Grab Bag (12 answers total)

Best answer: i think you would get a powerlaw by reducing the quincunx to a diagonal line of pins (say top left to bottom right) and dropping the ball in top left.

seems like poisson should be possible too by combining the quincunx with a conveyor belt that collects balls at intervals (assuming balls drop regularly). but i don't have exact details.

sorry, just picking off the easy ones...
posted by andrewcooke at 11:17 AM on July 31, 2015

For cauchy, coulnd't you just adjust the angle of the tails? But then remember Cauchy has no mean.
posted by MisantropicPainforest at 11:46 AM on July 31, 2015

Response by poster: Yeah, of course technically we would need infinite boxes for the Cauchy but this is also true for the Gaussian, just not too relevant because of the thin tail.

Any concrete or online examples or citations towards examples?
posted by curuinor at 12:09 PM on July 31, 2015

Best answer: Since much of the natural world obeys stacastic laws, there are plenty of demonstrations that illustrate them. I remember one in which pins or nails were dropped on a board, but I don't remember what was being demonstrated.

Consider a planar mirror spinning so that its flat face is contuously facing a new direction. (Mirror on edge rotating around a vertical axis) If you have a laser pointed at the mirror, the reflected light will fall in a line on a wall behind the laser. The amount of light that falls on any point is distributed according to a heavy tailed distribution, possibly Cauchy.
posted by SemiSalt at 2:12 PM on July 31, 2015

Buffon's Needle is what I was thinking of above. It yields a value of pi through a statistical argument.
posted by SemiSalt at 2:21 PM on July 31, 2015

Best answer: Hardware-based random number generators would be machines that generate random numbers via thermal noise or other methods. This type of machine generates (in theory) identically uniformly distributed independent random bits, which can be transformed into other distributions (lognormal, etc.) via an inversion transform. Here's a paper that discusses that approach with more rigor. You can even use atmospheric turbulence as a machine for generating a source of these bits.
posted by a lungful of dragon at 2:22 PM on July 31, 2015

Best answer: According to the Witch of Agnesi article on Wikipedia, the cross section of a single wave on water and the shape of a smooth hill are both very well approximated by a Cauchy distribution, so I would guess that you might need only to remove the baffles between the slots at the bottom of the Q machine and allow it to fill with balls to the point there are several deep at the sides in order to get a pretty good Cauchy distribution.
posted by jamjam at 3:53 PM on July 31, 2015

Best answer: In other words and more narrowly, you might get a Cauchy distribution out of a Normal distribution formed out of slightly sticky particles and then allowed to
slump under the influence of gravity.
posted by jamjam at 6:42 PM on July 31, 2015

Best answer: This page appears to describe the light experiment that I talked about above.

Another article recasts the experiment in a quainter way positing a blind archer.

This site has an example of using dye molecules to illustrate the Poisson Distribution.


Computer programs being much quicker to make than physical devices, I think that computer models have pretty much replaced physical ones for this sort of thing.
posted by SemiSalt at 6:44 AM on August 1, 2015

Response by poster: Cool beans!

Yeah, I'm not constructing anything for the time being. But models are cool!
posted by curuinor at 9:27 AM on August 1, 2015

Can't you just invert the pick-a-model-and-apply-it-to-data paradigm? Instead of having data, then picking a parametric likelihood function that approximates the real life data generating process, just do the opposite. Meaning that since car accidents are usually modelled by a poisson distribution, can't car accidents in a specific intersection be the 'machine' that models a theoretical likelihood function?
posted by MisantropicPainforest at 9:34 AM on August 1, 2015

Response by poster: MisantropicPainforest: this Physica A paper will be of interest (paywall, but the abstract will be a good enough summary). It would be nicer to have something theoretically nicer, is all.
posted by curuinor at 12:59 PM on August 2, 2015 [1 favorite]