(Exclusively published in the Personal Journal of Shalsoh B. Ekhad and Doron Zeilberger)

Posted: Aug. 25, 2010.

One of the main uses of computers is to do statistical analysis of data. But, so far, the theory of statistics, and its noble mother, Probability theory, were all discovered and developed by lowly humans. No more! Computers can also develop probability theory (and statistics), and discover (and prove!) general theorems of much larger depth than those discovered by human-kind. Of course, at this time of writing, they still need these inferior humans to give them a head-start by teaching (i.e. programming) them to develop probability theory ab initio (only better!), but even this will soon be superfluous.

Sample Input and Output for the Maple package HISTABRUT

• If you toss a fair coin n times, and win a dollar if it lends Heads and lose a dollar if it lends Tails, in order to find the mean, variance, and an explicit formula for the normailzed even moments α_2r to order 2 (i.e. O(1/n³)) (of course the odd moments are zero),
  the input gives the output.

• If you toss a loaded coin n times, and win a dollar with probality 1/3 and lose a dollar with probability 2/3, in order to find the mean, variance, and an explicit formulas for the normailzed even moments α_2r, and normalized odd moments α_2r+1, to order 2 (i.e. O(1/n³)) ,
  the input gives the output.

• You roll a fair (cubic) die n times, and for each roll you win as many shekels as the number of dots it landed on, and consider the random variable: "total number of shekels won". The mean, variance, and an explicit formula for the normailzed even moments α_2r to order 2 (i.e. O(1/n³)) (of course the odd moments are zero),
  the input gives the output.

• Suppose you toss, n times, a loaded three-faced coin with prob. 1/2 of winning 3 dollars, 1/4 of winning one dollar and 1/4 of losing 4 dollars. Consider your total win. In order to find the mean, variance, and an explicit formulas for the normailzed even moments α_2r, and normalized odd moments α_2r+1, to order 2 (i.e. O(1/n³)) ,
  the input gives the output.

• If you are wondering about the statistical behavior of the random variable "number of three consectutive Heads" (that may or may not be parts of longer Head-runs), upon tossing a fair coin n times,
  the input gives the output.

• If you are wondering about the statistical behavior of the random variable "number of three consectutive Heads" (that are NOT part of a longer Head-run, i.e. the number of occurrences of the string THHHT), upon tossing a fair coin n times,
  the input gives the output.

• To get a 5-second proof of the asymptotic normality of the random variable "number of inversions" in a random n-permutation (first proved by Feller (vol. I, X.6)), with first-order asymptotics for the normalized moments to boot,
  the input gives the output.

• To get a proof of the asymptotic normality of the random variable "number of inversions" in a (uniform) random word with n 1's and n 2's, as well as a first-order finer asymptotics for the normalized moments
  the input gives the output.

• For information about the asymptotics of the discrete probability distribution given by the q-Catalan polynomials
  (1-q)(q)_2n/(q)_n²/(1-qⁿ⁺¹)
  the input gives the output.

• To prove that the "continuous Stanley distribution" is the same as the normal distribution, but also to get a more refined asymptotics
  the input gives the output.

  (Note: see Shalosh B. Ekhad's note.)

• To get beautiful pictures of the limiting distribution of the random variable "duration of the game" in a gambler's ruin game that starts at the middle, or one third from either end,
  the input gives the output.
  (Notice the fat tails!)

• Suppose you walk in the plane, and with prob. 1/3 you stay in place, and with prob. 1/3 each you go to the left or up, and if you consider the bi-variate distibution (number of horizontal steps taken, number of vertical steps taken) after n steps. To get the means, variances, and the (normalized) mixed moments (up to the 10-th order)
  the input gives the output.

• Conisder the bi-variate distribution (inv,maj) defined on n-permutations. To find the means, variances and, more interestingly, the normalized mixed moments, up to the fourth-order
  the input gives the output.
  (Note that this seems to indicate that these are joint-independently asymptotically normal. A rigorous proof is coming up by Andrew Baxter and Doron Zeilberger)

• To get the means, variances, and mixed moments (up to the 8th) for the pair of random variables "number of two consecutive Heads" and "number of two consecutive Tails" upon tossing a fair coin n times,
  the input gives the output.

• To get the means, variances, and mixed moments (up to the 8th) for the pair of random variables "number of HHTH" and "number of TTHT" upon tossing a fair coin n times,
  the input gives the output.

• To (empirically, symbolically) check that the bivariate pair of r.v. (number of occurrences of HTT, number of occurrences of THH), in tossing a fair coin n times, as n goes to infinity, tends to a pair of correlated normal variables (with the apropriate correlation),
  the input gives the output.

• To get a verbose account of the mean, variance, correlation, and first few mixed moments for the pair of r.v. (Number of HT, Number of TH),
  the input gives the output.

• To get the means, variances, and most interestingly, the bi-variate moment-(exponential)generating-function (i.e. inverse Fourier Transform) of the limiting distribution of the sequence of discrete prob. distributions whose prob. generating functions is ((1+x1+x2)/3)ⁿ,
  the input gives the output.

• To get the means, variances, and most interestingly, the tri-variate moment-(exponential)generating-function (i.e. inverse Fourier Transform) of the limiting distribution of the sequence of tri-variate discrete prob. distributions whose prob. generating functions is (1+x[1]+x[2]+x[3]+2*x[1]*x[2]*x[3])/6)ⁿ
  the input gives the output.

• To get the mixed-moments for the famous pair (NumberOfInversions,MajorIndex) defined on n-permutations up to order 10 (except that it was unable to find the (10,10)-mixed moment), using the previously computed first 30 terms (gotten from procedure GnD), and saved in the file oGnD30,
  the input gives the output.

• To guess from the above output EXPLICIT expression for the asymptotics (to order 1) of the famous pair (NumberOfInversions,MajorIndex), proving the Baxter-Zeilberger deep theorem that they are asymptotically joint-independently-asymptotically-normal,
  the input gives the output.

Doron Zeilberger's Home Page