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 http://cs.smith.edu/~orourke/MathOverflow/GaltonBoard.jpg
(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 http://cs.smith.edu/~orourke/MathOverflow/HandFaceImprint.jpg
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...

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...

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 http://cs.smith.edu/~orourke/MathOverflow/GaltonBoard.jpg
(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 http://cs.smith.edu/~orourke/MathOverflow/HandFaceImprint.jpg
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...

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 http://cs.smith.edu/~orourke/MathOverflow/GaltonBoard.jpg
(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 http://cs.smith.edu/~orourke/MathOverflow/HandFaceImprint.jpg
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...

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 http://cs.smith.edu/~orourke/MathOverflow/GaltonBoard.jpg
(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 http://cs.smith.edu/~orourke/MathOverflow/HandFaceImprint.jpg
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...

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 http://cs.smith.edu/~orourke/MathOverflow/GaltonBoard.jpg
(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 http://cs.smith.edu/~orourke/MathOverflow/HandFaceImprint.jpg
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...