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...

anymathematician's face aside from, perhaps, Newton? $\endgroup$2more comments