Zeno -- a collection of helper functions for symbolic computations in Mathematica

Purpose of the project

In my recent research, I found that I often have to manipulate large symbolic expressions. Sometimes these equations are even hard to copy by hand on paper. Using Mathematica helped a lot. It provides many powerful features for manipulate symbolic expressions--when I do not know how to transform an expression through programming, at least I can copy-paste it and change it manually.

Zeno is a collection of helper functions that I wrote the past few months. Usually they are just wrappers for one-line codes. But I gradually learned to write them in better and more conciser ways, thanks to many helps that I got from mathematica.stackexchange.com. Hopefully these will also be helpful for others. At least by checking the code, you will see how to do certain things in Mathematica.

Zeno currently contains these packages:

• Zeno.m -- helper functions for manipulating symbolic expressions.
• ChernoffBound.m -- concentration inequalities.
• CalcMoment.m -- formulas for calculating moments of random variables.

(The project is named after the accent Greek philosopher Zeno of Citium.)

Usage

Currently these packages do not have dependency. So you can simply download a package, say CalcMoment.m, in the same folder as your Mathematica notebook and load it by

SetDirectory[NotebookDirectory[]];
Needs["CalcMoment`"]

You can also put it in FileNameJoin@{$UserBaseDirectory, "Applications"} and then load it in your notebook by

Needs["CalcMoment`"]

To see available functions, type

?CalcMoment`*

Check this test notebook for examples of how to use these packages.

ChernoffBound.m -- Chernoff's bounds

Chernoff's bounds are some concentration inequalities widely used in probabilistic combinatorics. ProbChopper.m includes several versions of Chernoff's bounds for binomial distributions. To see all available inequalities available, type

ChernoffPrintAll[]

The proofs of these inequalities can be found in

1. Michael Molloy and Bruce Reed, Graph Coloring and Probabilistic Methods, Springer Science & Business Media, 29 Jun 2013.

2. Michael Mitzenmacher, Eli Upfal, Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press, 31 Jan 2005.

3. Noga Alon, Joel H. Spencer, The Probabilistic Method, 2nd ed., John Wiley & Sons, 5 Apr 2004.

4. Wikipedia -- Chernoff Bound

CalcMoment.m -- Truncated Moments

Given a random variable Y, sometimes we want to compute the expectation of Y[Y < a], where [P]=1 if the P and true and [P]=0 if P is false. (The notation is called Iverson Bracket.)

In Patrice, it can be convenient to write E(Y[Y < a]) in terms of the left and right tails of $Y$. These are two function TruncatedMoment and TruncatedExpMoment contained in CalcMoment.m which does this. You can load it and try it in a Mathematica notebook like this

SetDirectory[NotebookDirectory[]];
Get["CalcMoment`"]

gdist = GammaDistribution[3, 7]
gtest = TruncatedMoment[gdist, p, a] == TruncatedMoment[gdist, p, a, MomentForm -> "Left"] == TruncatedMoment[gdist, p, a, MomentForm -> "Right"]
Assuming[0 < a && p >= 1, gtest // Activate // FullSimplify]

edist = ExponentialDistribution[1]
etest = TruncatedExpMoment[edist, a] == TruncatedExpMoment[edist, a, MomentForm -> "Left"] == TruncatedExpMoment[edist, a, MomentForm -> "Right"]
Assuming[0 < a, etest // Activate // FullSimplify]