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odstranena zbytecna veta

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Petr Velička 2025-03-23 14:17:14 +01:00
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stredni-hodnota.tex
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@ -346,7 +346,6 @@ Následující věta nám umožňuje jednoznačně popisovat rozdělení jak pod
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
Idea je taková, že postupně budeme rozdělovat integrály a odhadovat hodnoty těchto omezených integrálů.
Mějme $T \in \R$ a $a < b$, potom Mějme $T \in \R$ a $a < b$, potom
$$ \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \varphi_X(t) dt = \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \int_\R e^{itx} dP_X = $$ $$ \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \varphi_X(t) dt = \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \int_\R e^{itx} dP_X = $$
$$ \int_{-T}^T \int_{-\infty}^{\infty} \frac{e^{it(x - a)} - e^{it(x - b)}}{2\pi it} dP_Xdt \overset{Fubini}{=} \int_{-\infty}^\infty \int_{-T}^T \frac{e^{it(x - a)} - e^{it(x - b)}}{2\pi it} dtdP_X. $$ $$ \int_{-T}^T \int_{-\infty}^{\infty} \frac{e^{it(x - a)} - e^{it(x - b)}}{2\pi it} dP_Xdt \overset{Fubini}{=} \int_{-\infty}^\infty \int_{-T}^T \frac{e^{it(x - a)} - e^{it(x - b)}}{2\pi it} dtdP_X. $$