Commit d445af83 authored by Parth Mittal's avatar Parth Mittal
Browse files

add a macro for Var, and use it

parent defbcae6
......@@ -189,11 +189,11 @@ is $2$-independent). Hence $\E[X_{r, j}] = 1 / 2^r$. By linearity of
expectation, $\E[Y_{r}] = d / 2^r$.
We will also use the variance of these variables -- note that
$${\rm Var}[X_{r, j}] \leq \E[X_{r, j}^2] = \E[X_{r, j}] = 1/2^r$$
$$\Var[X_{r, j}] \leq \E[X_{r, j}^2] = \E[X_{r, j}] = 1/2^r$$
And because $h$ is $2$-independent, the variables $X_{r, j}$ and $X_{r, j'}$
are independent for $j \neq j'$, and hence:
$${{\rm Var}}[Y_{r}] = \sum_{j : f_j > 0} {\rm Var}[X_{r, j}] \leq d / 2^r $$
$$\Var[Y_{r}] = \sum_{j : f_j > 0} \Var[X_{r, j}] \leq d / 2^r $$
Now, let $a$ be the smallest integer such that $2^{a + 1/2} \geq 3d$. Then we
have:
......@@ -208,7 +208,7 @@ $$ \Pr[\hat{d} \leq d / 3] = \Pr[ Y_{b + 1} = 0] \leq
\Pr[ \vert Y_{b + 1} - \E[Y_{b + 1}] \vert \geq d / 2^{b + 1} ]$$
Using Chebyshev's inequality, we get:
$$ \Pr[\hat{d} < d / 3] \leq {{\rm Var}[Y_b] \over (d / 2^{b + 1})^2} \leq
$$ \Pr[\hat{d} < d / 3] \leq {\Var[Y_b] \over (d / 2^{b + 1})^2} \leq
{2^{b + 1} \over d} \leq {\sqrt{2} \over 3}$$
\qed
......
......@@ -170,6 +170,7 @@
\def\E{{\bb E}}
\def\Pr{{\rm Pr}\mkern0.5mu}
\def\Prsub#1{{\rm Pr}_{#1}}
\def\Var{{\rm Var}\mkern0.5mu}
% Vektory
\def\t{{\bf t}}
......
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