From cb35d82080a5aa819c6a3f0ca97a1846bbe0b278 Mon Sep 17 00:00:00 2001
From: Aaron Huber <ahuber@buffalo.edu>
Date: Wed, 12 Jun 2019 12:03:01 -0400
Subject: [PATCH] More detail to Chebyshev Inequality derivation

---
 analysis.tex | 10 +++++++---
 macros.tex   |  1 +
 2 files changed, 8 insertions(+), 3 deletions(-)

diff --git a/analysis.tex b/analysis.tex
index 0b87631..36485d0 100644
--- a/analysis.tex
+++ b/analysis.tex
@@ -115,14 +115,18 @@ By \eqref{eq:variance} we have then
 \sd &< 2^N\big(\sqrt{\frac{2\prob^2}{\sketchCols}}\big)\\
 \sdRel& < \sqrt{\frac{2}{\sketchCols}}.
 \end{align*}
-Since the sketch has multiple trials, a probability of exceeding error bound smaller than one half guarantees an estimate that is less than or equal to the error bound when taking the median of all trials.  Expressing this as Chebyshev's Inequality yields
+Since the sketch has multiple trials, a probability of exceeding error bound smaller than one half guarantees an estimate that is less than or equal to the error bound when taking the median of all trials.  Expressing this in Chebyshev's Inequality yields
 \begin{equation*}
 \cheby.
 \end{equation*}
 Substituting $\mu\epsilon$ for $k\sd$ and solving for $\sketchCols$ results in
 \begin{align*}
- \chebyK\Rightarrow& \\
-&\sketchCols = \frac{6}{\mu^2\epsilon^2}
+&k\sdRelVar = \mu\epsilon\\
+&k = \frac{\mu\epsilon}{\sdRelVar}\\
+&k = \frac{\mu\epsilon\sqrt{\sketchCols}}{\sqrt{2}}\\
+&k^2 = \frac{1}{\big(\frac{\mu\epsilon\sqrt{\sketchCols}}{\sqrt{2}}\big)^2}\\
+&k^2 = \frac{2}{\big(\mu^2\epsilon^2\sketchCols}\\
+&\chebyK\Rightarrow \sketchCols = \frac{6}{\mu^2\epsilon^2}
 \end{align*}
 
 
diff --git a/macros.tex b/macros.tex
index ed39bef..6e7ec3f 100644
--- a/macros.tex
+++ b/macros.tex
@@ -61,6 +61,7 @@
 \newcommand{\varParam}[1]{Var\bigParamBox{#1}}
 \newcommand{\varSym}{\sd^2}
 \newcommand{\sd}{\sigma}
+\newcommand{\sdRelVar}{\sqrt{frac{2}{\sketchCols}}}
 \newcommand{\sdRel}{\sd_{rel}}
 %%%%%%%%%%%%%%%%%
 %Chebyshev