From 28bc88cbc5a6898971e5bdea49c7e0cbc981aac9 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Mon, 17 Jul 2023 12:36:38 +0200 Subject: [PATCH] fixed chebyshev --- inputs/lecture_16.tex | 1 + inputs/prerequisites.tex | 4 ++-- 2 files changed, 3 insertions(+), 2 deletions(-) diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index 4195844..dca129c 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -197,6 +197,7 @@ Typically $\cF_n = \sigma(X_1, \ldots, X_n)$ for a sequence of random variables. \footnote{In this form it means, that there is some filtration, that we don't explicitly specify}. Then $(f(X_n))_n$ is a sub-martingale. + Likewise, if $f$ is concave, then $((f(X_n))_n$ is a super-martingale. \end{corollary} \begin{proof} Apply \autoref{cjensen}. diff --git a/inputs/prerequisites.tex b/inputs/prerequisites.tex index b78ea23..cdad15c 100644 --- a/inputs/prerequisites.tex +++ b/inputs/prerequisites.tex @@ -280,7 +280,7 @@ This is taken from section 6.1 of the notes on Stochastik. Let $X$ be a random variable and $a > 0$. Then \[ - \bP[|X - \bE(X)| \ge a] \le \frac{\Var(X)}{a}. + \bP[|X - \bE(X)| \ge a] \le \frac{\Var(X)}{a^2}. \] \end{theorem} \begin{proof} @@ -288,7 +288,7 @@ This is taken from section 6.1 of the notes on Stochastik. \begin{IEEEeqnarray*}{rCl} \bP[|X-\bE(X)| \ge a] &=& \bP[|X - \bE(X)|^2 \ge a^2]\\ - &\overset{\text{Markov}}{\ge}& \frac{\bE[|X - \bE(X)|^2]}{a^2}. + &\overset{\text{Markov}}{\le}& \frac{\bE[|X - \bE(X)|^2]}{a^2}. \end{IEEEeqnarray*} \end{proof}