% Lecture 1 - 2023-04-04

First, let us recall some basic definitions:
\begin{definition}
    A \vocab{probability space} is a triplet $(\Omega, \cF, \bP)$,
    such that
    \begin{itemize}
        \item $\Omega \neq \emptyset$,
        \item $\cF$ is a $\sigma$-algebra over $\Omega$, i.e.~$\cF \subseteq \cP(\Omega)$ and
            \begin{itemize}
                \item $\emptyset, \Omega \in \cF$,
                \item $A \in \cF \implies A^c \in \cF$,
                \item $A_1, A_2,\ldots \in \cF \implies \bigcup_{i \in \N} A_i \in \cF$.
            \end{itemize}
            The elements of $\cF$ are called \vocab[Event]{events}.
        \item $\bP$ is a \vocab{probability measure}, i.e.~$\bP$ is a function $\bP: \cF \to [0,1]$
            such that
            \begin{itemize}
                \item $\bP(\emptyset) = 1$, $\bP(\Omega)  = 1$,
                \item $\bP\left( \bigsqcup_{n \in \N} A_n \right) = \sum_{n \in \N} \bP(A_n)$
                    for mutually disjoint $A_n \in \cF$.
            \end{itemize}
    \end{itemize}
\end{definition}


\begin{definition}
    A \vocab{random variable} $X : (\Omega, \cF) \to (\R, \cB(\R))$
    is a measurable function, i.e.~for all $B \in \cB(\R)$ we have $X^{-1}(B) \in \cF$.
    (Equivalently $X^{-1}\left( (a,b] \right) \in \cF$ for all $a < b \in \R$ ).
\end{definition}

\begin{definition}
    $F: \R \to \R_+$ is a \vocab{distribution function} iff
    \begin{itemize}
        \item $F$ is monotone non-decreasing,
        \item $F$ is right-continuous,
        \item  $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$.
    \end{itemize}
\end{definition}
\begin{fact}
    Let $\bP$ be a probability measure on $(\R, \cB(\R))$.
    Then $F(x) \coloneqq\bP\left( (-\infty, x] \right)$
    is a probability distribution function.
    (See lemma 2.4.2 in the lecture notes of Stochastik)
\end{fact}
The converse to this fact is also true:
\begin{theorem}[Kolmogorov's existence theorem / basic existence theorem of probability theory]
    \label{kolmogorovxistence}
    Let $\cF(\R)$ be the set of all distribution functions on $\R$
    and let $\cM(\R)$ be the set of all probability measures on $\R$.
    Then there is a one-to-one correspondence between $\cF(\R)$ and $\cM(\R)$
    given by
    \begin{IEEEeqnarray*}{rCl}
        \cM(\R)  &\longrightarrow & \cF(\R)\\
        \bP &\longmapsto & \begin{pmatrix*}[l]
            \R &\longrightarrow & \R_+ \\
            x &\longmapsto & \bP((-\infty, x]).
        \end{pmatrix*}
    \end{IEEEeqnarray*}
\end{theorem}
\begin{proof}
    See theorem 2.4.3 in Stochastik.
\end{proof}
\begin{example}[Some important probability distribution functions]\hfill
    \begin{enumerate}[(1)]
        \item \vocab{Uniform distribution} on $[0,1]$:
            \[
                F(x) = \begin{cases}
                    0 & x \in (-\infty, 0],\\
                    x & x \in (0,1],\\
                    1 & x \in (1,\infty).\\
                \end{cases}
            \]
        \item \vocab{Exponential distribution}:
            \[
            F(x) = \begin{cases}
                1 - e^{-\lambda x} &  x \ge 0,\\
                0 & x < 0.
            \end{cases}
            \]
        \item \vocab{Gaussian distribution}:
            \[
            \Phi(x) \coloneqq \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{y^2}{2}} dy.
            \]
        \item $\bP[X = 1] = \bP[X = -1] =  \frac{1}{2}$ :
            \[
            F(x) = \begin{cases}
                0 & x \in (-\infty, -1),\\
                \frac{1}{2} & x \in [-1,1),\\
                1 & x \in [1, \infty).
            \end{cases}
            \]
    \end{enumerate}
\end{example}