123 lines
4.6 KiB
TeX
123 lines
4.6 KiB
TeX
% Lecture 1 - 2023-04-04
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First, let us recall some basic definitions:
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\begin{definition}
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A \vocab{probability space} is a triplet $(\Omega, \cF, \bP)$,
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such that
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\begin{itemize}
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\item $\Omega \neq \emptyset$,
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\item $\cF$ is a $\sigma$-algebra over $\Omega$, i.e.~$\cF \subseteq \cP(\Omega)$ and
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\begin{itemize}
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\item $\emptyset, \Omega \in \cF$,
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\item $A \in \cF \implies A^c \in \cF$,
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\item $A_1, A_2,\ldots \in \cF \implies \bigcup_{i \in \N} A_i \in \cF$.
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\end{itemize}
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The elements of $\cF$ are called \vocab[Event]{events}.
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\item $\bP$ is a \vocab{probability measure}, i.e.~$\bP$ is a function $\bP: \cF \to [0,1]$
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such that
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\begin{itemize}
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\item $\bP(\emptyset) = 1$, $\bP(\Omega) = 1$,
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\item $\bP\left( \bigsqcup_{n \in \N} A_n \right) = \sum_{n \in \N} \bP(A_n)$
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for mutually disjoint $A_n \in \cF$.
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\end{itemize}
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\end{itemize}
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\end{definition}
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\begin{definition}
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A \vocab{random variable} $X : (\Omega, \cF) \to (\R, \cB(\R))$
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is a measurable function, i.e.~for all $B \in \cB(\R)$ we have $X^{-1}(B) \in \cF$.
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(Equivalently $X^{-1}\left( (a,b] \right) \in \cF$ for all $a < b \in \R$ ).
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\end{definition}
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\begin{definition}
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$F: \R \to \R_+$ is a \vocab{distribution function} iff
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\begin{itemize}
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\item $F$ is monotone non-decreasing,
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\item $F$ is right-continuous,
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\item $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$.
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\end{itemize}
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\end{definition}
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\begin{fact}
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Let $\bP$ be a probability measure on $(\R, \cB(\R))$.
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Then $F(x) \coloneqq\bP\left( (-\infty, x] \right)$
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is a probability distribution function.
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(See lemma 2.4.2 in the lecture notes of Stochastik)
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\end{fact}
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The converse to this fact is also true:
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\begin{theorem}[Kolmogorov's existence theorem / basic existence theorem of probability theory]
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\label{kolmogorovxistence}
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Let $\cF(\R)$ be the set of all distribution functions on $\R$
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and let $\cM(\R)$ be the set of all probability measures on $\R$.
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Then there is a one-to-one correspondence between $\cF(\R)$ and $\cM(\R)$
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given by
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\begin{IEEEeqnarray*}{rCl}
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\cM(\R) &\longrightarrow & \cF(\R)\\
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\bP &\longmapsto & \begin{pmatrix*}[l]
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\R &\longrightarrow & \R_+ \\
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x &\longmapsto & \bP((-\infty, x]).
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\end{pmatrix*}
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\end{IEEEeqnarray*}
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\end{theorem}
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\begin{proof}
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See theorem 2.4.3 in Stochastik.
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\end{proof}
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\begin{example}[Some important probability distribution functions]\hfill
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\begin{enumerate}[(1)]
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\item \vocab{Uniform distribution} on $[0,1]$:
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\[
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F(x) = \begin{cases}
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0 & x \in (-\infty, 0],\\
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x & x \in (0,1],\\
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1 & x \in (1,\infty).\\
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\end{cases}
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\]
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[samples=1000, xmin=-1, xmax=2, width=10cm, height=5cm]
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\addplot[] {and(x>0,x<=1) * x + (x>1)};
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\item \vocab{Exponential distribution}:
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\[
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F(x) = \begin{cases}
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1 - e^{-\lambda x} & x \ge 0,\\
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0 & x < 0.
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\end{cases}
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\]
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[samples=1000, smooth, width=10cm, height=5cm, xmin=-2, xmax=5]
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\addplot[] {(x > 0) * (1 - exp( - 5 * x))};
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\item \vocab{Gaussian distribution}:
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\[
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\Phi(x) \coloneqq \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{y^2}{2}} dy.
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\]
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\item $\bP[X = 1] = \bP[X = -1] = \frac{1}{2}$ :
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\[
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F(x) = \begin{cases}
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0 & x \in (-\infty, -1),\\
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\frac{1}{2} & x \in [-1,1),\\
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1 & x \in [1, \infty).
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\end{cases}
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\]
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[samples=1000, width=10cm, height=5cm]
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\addplot[] {and(x >= -1, x < 1) * 0.5 + (x >= 1)};
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\end{enumerate}
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\end{example}
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