fixed typo

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Josia Pietsch 2023-07-19 22:47:29 +02:00
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commit f60b0a0bad
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2 changed files with 37 additions and 3 deletions

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@ -141,7 +141,7 @@ We have
Then
\begin{itemize}
\item $\phi_{a X + b}(t) = e^{\i t b} \phi_X(\frac{t}{a})$,
\item $\phi_{X + Y}(t) = \phi_X(t) + \phi_Y(t)$.
\item $\phi_{X + Y}(t) = \phi_X(t) \cdot \phi_Y(t)$.
\end{itemize}
\end{fact}
\begin{proof}

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@ -48,6 +48,7 @@ from the lecture on stochastic.
\pagebreak
\begin{theorem}+
% \footnote{see exercise 3.4}
\label{thm:convergenceimplications}
\vspace{10pt}
Let $X$ be a random variable and $X_n, n \in \N$ a sequence of random variables.
@ -247,7 +248,20 @@ from the lecture on stochastic.
We have $[-k,k] \uparrow \R$, hence $\mu([-k,k]) \uparrow \mu(\R)$.
\end{proof}
\begin{theorem}[Riemann-Lebesgue]
\begin{theorem}+[Change of variables formula]
Let $X$ be a random variable with a continuous density $f$,
and let $g: \R \to \R$ be continuous such that
$g(X)$ is integrable.
Then
\[
\bE[g(X)] = \int g \circ X \dif \bP
= \int_{-\infty}^\infty g(y) f(y) \lambda(\dif y)
= \int_{-\infty}^\infty g(y) f(y) \dif y.
\]
\end{theorem}
\begin{theorem}+[Riemann-Lebesgue]
%\footnote{see exercise 3.3}
\label{riemann-lebesgue}
Let $f: \R \to \R$ be integrable.
Then
@ -256,7 +270,27 @@ from the lecture on stochastic.
\]
\end{theorem}
\begin{theorem}+[Fubini-Tonelli]
%\footnote{exercise sheet 1}
Let $(\Omega_{i}, \cF_i, \bP_i), i \in \{0,1\}$
be probability spaces and $\Omega \coloneqq \Omega_0 \otimes \Omega_1$,
$\cF \coloneqq \cF_1 \otimes\cF_2$,
$\bP \coloneqq \bP_0 \otimes \bP_1$.
Let $f \ge 0$ be $(\Omega, \cF)$-measurable,
then
\[
\Omega_0 \ni x \mapsto \int_{\Omega_{2}} f(x,y) \bP_2(\dif y)
\]
and
\[
\Omega_1 \ni y \mapsto \int_{\Omega_1} f(x,y) \bP_1(\dif x)
\]
are measurable, and
\[
\int f \dif \bP = \int_{\Omega_1} \int_{\Omega_2} f(x,y) \bP_2(\dif y) \bP_1(\dif x)(\dif x)
= \int_{\Omega_2} \int_{\Omega_1} f(x,y) \bP_1(\dif x) \bP_2(\dif y).
\]
\end{theorem}
\subsection{Inequalities}
This is taken from section 6.1 of the notes on Stochastik.