# Ping-pong relief map of a given function $z=f(x,y)$

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:

(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.

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