HISTABRUT: A Maple Package for Symbol-Crunching in Probability theory

By Doron Zeilberger

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

Posted: Aug. 25, 2010.

Last update of this webpage (but not article): Jan. 3, 2016.

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.

HISTABRUT, the Maple package that this article describes.

[Some browsers need the .txt extension here is the same package with .txt HISTABRUT.txt]

Sample Input and Output for the Maple package HISTABRUT

In a very interesting article linking lambda calculus to planar maps. Noam Zeilberger came up with a sequence of polynomials P_n(x), whose ordinary generating function, with respect to z, satisfies the functional-differential equation (Eq. (6), p. 12) there, Lind(z,x) (let's call if f(z,x))

f(z,x)=x+z*(f(z,x)-f(z,0))^2 + z*diff(f(z,x),x)

Here x is the catalytic variable, and the main interest is the sequence of numbers P_n(0), in other words, the coefficients of f(z,0).

But this also raises the question of whether the random variable accounted for by the exponent of x, is asympotically normal. Using the first 300 terms, it seems that it is.

In order to crank-out this sequence we the maple code


yields the following input and output files

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