This link http://mathworld.wolfram.com/GaltonBoard.html suggests that a certain specific placement of pegs can be used to simulate a binomial or a normal distribution. Is there a specific peg configuration that can be used to generate the uniform distribution. In general, can any standard distribution be simulated using configuration of pegs?
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1$\begingroup$ uniform distribution: just place the pegs in a rectangular area (rather than in a triangular area), as the ratio height/width increases the distribution approaches the uniform one. $\endgroup$– Carlo BeenakkerCommented Feb 27, 2018 at 20:56
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1$\begingroup$ @CarloBeenakker how can one obtain a uniform distribution among the slots if the balls all start in the center? Maybe I am missing something, but it seems the balls basically undergo a random walk with initial distribution concentrated at a point. $\endgroup$– Nawaf Bou-RabeeCommented Feb 27, 2018 at 21:09
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3$\begingroup$ it's a random walk in the $x$-direction with reflecting boundaries in the $y$-direction, say the board occupies the region between $-W/2<y<W/2$; the balls start initially at $y=0$, then they spread out, and after they have moved a length $L\gg W$ in the $x$-direction their density along $y$ is uniform. $\endgroup$– Carlo BeenakkerCommented Feb 27, 2018 at 21:16
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1$\begingroup$ See whether probability.ca/jeff/ftpdir/quincunx.pdf, particularly Section 6, answers your question. $\endgroup$– Gerry MyersonCommented Feb 27, 2018 at 22:23
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1$\begingroup$ Once you have a uniform distribution, via Carlo's method or any other, you can take appropriate ranges of the interval and distribute them uniformly across other intervals of varying sizes. In this way you can simulate any distribution you like to any desired accuracy. A somewhat related result, in continuous time, is the Skorohod embedding theorem which says you can get any square-integrable distribution by stopping a Brownian motion. See also math.stackexchange.com/a/2106954/822. $\endgroup$– Nate EldredgeCommented Feb 27, 2018 at 22:53
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The method above to converge to the uniform distribution is in the right spirit to generalize the Galton board. A simple regular repeating pattern.
The following is much less aesthetic (and my crude illustration makes it worse) but does show how to get an exact uniform distribution, at least with $2^k$ cells.
answered Feb 28, 2018 at 7:17