Automated Derivation of Limiting Distributions Of Combinatorial Random Variables Whose Generating Functions are Rational
By Doron Zeilberger
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Written: Dec. 24, 2016
[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger]
In this seminal article
I showed how to teach a computer to discover (and prove!) Central Limit Theorems for a wide class
of combinatorial random variables. Here I extend it even further, to an even larger class, namely those
whose bivariate generating functions are rational functions in both variables.
The same methodology can be applied to get the multidistribution with several combinatorial random variables.
But its implementation would have to wait for the future.
Maple packages

BiVariateMoms.txt,
a Maple package to automatically derive explicit expressions for the expectation, variance, higher moments,
and limiting distributions of combinatorial random variables whose generating functions are given by bivariate generating functions.

RandComps.txt,
a Maple package that tests the above package, by drawing random compositions of an arbitrary alphabet, and
exploring the distribution of the random variable `number of occurrences of` one of its members.
Sample Input and Output Files for the Maple package BiVariateMoms.txt

If you want to see a computergenerated article about the average, variance, and the moments up to the 6th, of the
random variable "number of 2's" defined on the set of compositions of n, only using 1's and 2's
the
input file generates
output file.

If you want to see a computergenerated article about the average, variance, and the moments up to the 6th, of the
random variable "number of Heads" defined on the set outcomes of n cointosses of a fair coin
the
input file generates
output file.

If you want to see a computergenerated article about the average, variance, and the moments up to the 4th, of the
random variable "number of monomers" defined on the set of monomerdimer tilings of a 2 by n rectangle , 4 by n rectangle , and 6 by n
rectangle,
the
input file generates
output file.

If you want to see graphs of the function x vs. limit of log(a(n,n*x))/n as n goes to infinity,
where a(n,m) is the coefficient of t^{n}w^{m} in the Maclaurin expansion of the
rational function f(t,w), for various rational functions,
the
input file generates
output file.

If you want to see several computergenerated articles about the average, variance, and the moments up to the 6th, of the random variable

"number of occurences of 2s" defined on the set of compositions of n, only using {2,3}

"number of occurences of 3s" defined on the set of compositions of n, only using {2,3}

"number of occurences of 2s" defined on the set of compositions of n, only using {2,3,5}

"number of occurences of 3s" defined on the set of compositions of n, only using {2,3,5}

"number of occurences of 5s" defined on the set of compositions of n, only using {2,3,5}
the
input file generates
output file.
Sample Input and Output Files for the Maple package RandComp.txt

If you want to see the outcome of an experiment where 1000 (uniformlyatrandom) compositions of 100,
that only use {1,2}, and keeping track of the random variable `number of occurrences of 1'.
It also returns the exact expectation and standarddeviation (the last component)
the
input file generates
output file.

If you want to see the outcome of an experiment where 2000 (uniformlyatrandom) compositions of 200,
that only use {1,3,5}, and keeping track of the random variable `number of occurrences of 3'.
It also returns the exact expectation and standarddeviation (the last component)
the
input file generates
output file.

If you want to see the outcome of an experiment where 2000 (uniformlyatrandom) compositions of 200,
that only use {2,3,7}, and keeping track of the random variable `number of occurrences of 7'.
It also returns the exact expectation and standarddeviation (the last component)
the
input file generates
output file.
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