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[AW] Fix error in chebyshev inequality, fixes #7

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janishutz
2026-07-15 14:21:31 +02:00
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semester2/algorithms-and-probability/parts/probability/approximating-probabilities.tex
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@@ -15,7 +15,7 @@ Markov's inequality is fairly straight forward to prove, and it already allows u
\begin{theorem}[]{Chebyshev's inequality}
Let $\mathcal{X}$ be a random variable and $t > 0 \in\R$. Then we have
\begin{align*}
\Pr[|\mathcal{X} - \E[\mathcal{X}| \geq t]] \leq \frac{\text{Var}[\mathcal{X}]}{t^2} \Longleftrightarrow \Pr[|\mathcal{X} - \E[\mathcal{X}]| \geq t \cdot \sqrt{\text{Var}[\mathcal{X}]}] \leq \frac{1}{t^2}
\Pr[|\mathcal{X} - \E[\mathcal{X}]| \geq t] \leq \frac{\text{Var}[\mathcal{X}]}{t^2} \Longleftrightarrow \Pr[|\mathcal{X} - \E[\mathcal{X}]| \geq t \cdot \sqrt{\text{Var}[\mathcal{X}]}] \leq \frac{1}{t^2}
\end{align*}
\end{theorem}