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We are all familiar with the Galton Machine and the images of the balls cascading through the device and ending up in bins which ultimately show a likeness to the binomial distribution. Most everyone will agree that the balls exhibit random and unpredictable behavior. My question is twofold:

(1) Can the emergence of the balls into the binomial curve be considered a mathematical proof that the behavior of any single ball is unpredictable?

(2) Can the emergence of the binomial curve for any mathematical function be considered a mathematical proof that the elements which create the distribution are in fact unpredictable?

The very fluffy Panda
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william hird
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1 Answers1

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The answer on both accounts is No.

There are (deterministic!) algorithmns, called pseudo random number generators (PRNG) which create, well pseudo random numbers. These numbers in princple follow a deterministic pattern, but in pratice behave like truely random numbers. In particular, you can use a PRNG to simulate statistical experiments like throwing a dice, and for the right kind of experiment, the results will be binomially distributed.

fgp
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  • Could you cite any papers or literature to support your answer? – william hird Apr 17 '14 at 02:33
  • @williamhird I included a link to an wikipedia article, start reading there... – fgp Apr 17 '14 at 09:50
  • I read the article you linked but I didn't see any reference to how the PRNG was used in any experiment or algorithm that produced the binomial distribution. I did see a reference in Horowitz & Hill the Art of Electronics where they talk about a PRNG sequence being the equivalent of a random walk though... – william hird Apr 17 '14 at 14:30
  • @williamhird You can produce arbitrary distributions from a PRNG that yields pseudo-random numbers in $[0,1]$. The idea is to find the inverse $F^{-1}$ of the CDF $F$ of the distribution you want to generate, and set $X_n = F^{-1}(R_n)$ where $R_n$ are (pseudo-)random numbers in $[0,1]$. (If the CDF isn't invertible, one uses an appropriate pseudo-inverse) – fgp Apr 17 '14 at 22:26