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I have an idea to design a type of Galton's Board to "draw" a relief map of a given two-dimensional function $z=f(x,y)$. A typical Galton's Board drops, say, ping-pong balls through a series of evenly spaced pins into vertical bins to demonstrate that the balls distribute according to the binomial distribution, approximating the normal distribution:
           Galton's Board
(See this this link for an animation.)

First I would like to generalize this design to approximate an arbitrary function $y=f(x)$, which leads to my first question:

Q1. Which class of functions can be represented as a convex combination of normal distributions?

I know these functions are called mixture distributions, but I have not found a description of the total class representable. I am hoping that (say) any smooth function can be approximated.

Q2. Given a function $f(x)$ to approximate, how could one work backward to a pin distribution that would realize the approximation?

The result would be a type of user-designed Pachinko machine.

Q3. Can the above be generalized to two-dimensional functions $f(x,y)$?

Presumably the answer is Yes. If so, one could imagine a potentially mesmerizing Museum of Math display in which some famous visage emerges slowly as a ping-pong relief map.
           Imprint Toy
Q4. This final thought raises the question of which mathematician's face would be simultaneously most appropriate and most recognizable. :-) Sir Francis Galton is certainly appropriate...

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Nice question! But would people recognize any mathematician's face aside from, perhaps, Newton? –  Qiaochu Yuan Jul 6 '11 at 14:31
Most people won't even recognize Newton! –  drbobmeister Jul 6 '11 at 16:39
For Q1, there are restrictions beyond positivity even for smooth functions if you want equality. The tails of any convex combination of normal distributions do not drop too rapidly, so $\exp(-x^4)$ is not a convex combination of normal distributions. You might approximate it arbitrarily well, though. –  Douglas Zare Jul 6 '11 at 18:41
If you take a convolution of your favourite smooth function with a normal distribution with mean 0 and small standard deviation, then you get essentially your function back. This convolution though can be approximated by a finite mixture of normal distributions. –  Anthony Quas Jul 6 '11 at 20:23
PS: Hawking would be recognizable... –  Anthony Quas Jul 6 '11 at 20:24

1 Answer 1

This is a partial answer to Q2, and suggests to me that there is a physical arrangement which would give a yes answer to Q3.

If you use ordinary pins, you can probably get a dyadic approximation with some arrangement. Let me suggest using weighted pins as a partial solution, and then perhaps someone can implement a close enough approximation to a weighted pin with a series of dyadic pins.

So normalize things so that the function f has integral one over the interval [0,1], and is to be approximated by 2^k bins. Suppose p in [0,1] is the fraction of balls needed to represent the function on [0, 1/2], equivalently p is the integral of f from [0,1/2]. Then place a weighted pin very high such that it dumps p of the balls toward the pin over the interval [0, 1/2]. (You may want to put a divider right under this pin so that the ball doesn't jump to the [1/2,1] side.) Now recurse (k-1) more levels. Working backwards from this to get a horizontal arrangement should be clear, and of course one can use the physics of the situation to change the endpoints from dyadic rationals to something more appropriate to the desired function f.

It may be possible to emulate the bias by ever so slight horizontal adjustments of the pins, but you need to place the later pins just so that their bias accomodates the various trajectories of the incoming ball. But of course we have infinite precision pins and balls, so what's to worry?

Gerhard "Likes The Unreality of Mathematics" Paseman, 2011.07.06

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Also, I recommend the DVD Animusic 2. One of the pieces uses balls to do quite a performance. If this guy can do virtual music, certainly a virtual relief map of, e.g., Doron Zeilberger's left cheek is within reach. Gerhard "Likes The Alternate Realities, Too" Paseman, 2011.07.06 –  Gerhard Paseman Jul 6 '11 at 18:49

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