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