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probability-theory/inputs/a_2_additional_stuff.tex

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2023-07-07 17:42:38 +02:00
Important stuff not done in the lecture.
Moments:
$\bE[X^k]$
\begin{lemma}
Let $X, Y : \Omega \to [a,b]$
If $\bE[X^k] = \bE[Y^k]$,
for every $k \in \N_0$
then $X = Y$.
\end{lemma}
\begin{proof}
We have $\bE[p(X)] = \bE[p(Y)]$ for
every polynomial $p$.
Approximate $e^{\i t X}$
with polynomials and use Fourier transforms.
\end{proof}
Laplace transforms