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Zheyuan Wu
2024-12-02 11:07:52 -06:00
parent de70f0e9e8
commit 2f781e226a
2 changed files with 81 additions and 10 deletions
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pages/CSE347/CSE347_L11.md
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@@ -134,7 +134,7 @@ What we can do is to run the algorithm $x$ times independently with probability
The probability of the wrong decision is
$$
\binom{x}{\lfloor x/2\rfloor} \left(\frac{1}{2}-\epsilon\right)^{\lfloor x/2\rfloor}
\binom{x}{\lceil x/2\rceil} \left(\frac{1}{2}-\epsilon\right)^{\lceil x/2\rceil}
$$
I want to choose $x$ such that this is $\leq \delta$.
@@ -145,7 +145,7 @@ So
$$
\begin{aligned}
\binom{x}{\lfloor x/2\rfloor}\left(\frac{1}{2}-\epsilon\right)^{\lfloor x/2\rfloor}&\leq \left(\frac{xe}{x/2}\right)^{\lfloor x/2\rfloor}\left(\frac{1}{2}-\epsilon\right)^{-\lfloor x/2\rfloor\epsilon}
\binom{x}{\lceil x/2\rceil}\left(\frac{1}{2}-\epsilon\right)^{\lceil x/2\rceil}&\leq \left(\frac{xe}{x/2}\right)^{\lceil x/2\rceil}\left(\frac{1}{2}-\epsilon\right)^{-\lceil x/2\rceil\epsilon}
\end{aligned}
$$